|
%I #189 Feb 17 2022 00:13:34
%S 1,1,1,1,6,1,1,23,23,1,1,76,230,76,1,1,237,1682,1682,237,1,1,722,
%T 10543,23548,10543,722,1,1,2179,60657,259723,259723,60657,2179,1,1,
%U 6552,331612,2485288,4675014,2485288,331612,6552,1,1,19673,1756340,21707972,69413294,69413294,21707972,1756340,19673,1
%N Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (2*n - 2*k + 1)*T(n-1, k-1) + (2*k - 1)*T(n-1, k).
%C Rows are expansions of p(x,n) = 2^n*(1 - x)^(1 + n)*LerchPhi(x, -n, 1/2). Row sums are A000165. - _Roger L. Bagula_, Sep 16 2008
%C Eulerian numbers of type B. The n-th row of this triangle is the h-vector of the simplicial complex dual to a permutohedron of type B_(n-1). For example, the permutohedron of type B_2 is an octagon whose dual, also an octagon, has f-polynomial f(x) = 1 + 8*x + 8*x^2 and h-polynomial given by (x-1)^2 + 8*(x-1) + 8 = 1 + 6*x + x^2, giving [1,6,1] as row 3 of this table (see Fomin and Reading, p. 21). The corresponding triangle of f-vectors for the type B permutohedra is A145901. The Hilbert transform of the current array is A145905. - _Peter Bala_, Oct 26 2008
%C From _Peter Bala_, Oct 13 2011: (Start)
%C The row polynomials count the elements of the hyperoctahedral group B_n (the group of signed permutations on n letters) according to the number of type B descents (see Chow and Gessel).
%C Let P denote Pascal's triangle. Then the first column of the array P*(I-t*P^2)^(-1) (I the identity array) begins [1/(1-t),(1+t)/(1-t)^2,(1+6*t+t^2)/(1-t)^3,...]. The numerator polynomials are the row polynomials of this table. Similarly, in the array (I-t*A062715)^-1, the numerator polynomials in the first column produce the row polynomials of this table (but with an extra factor of t). Cf. A145901. (End)
%C The Dasse-Hartaut and Hitczenko paper (section 6.1.4) shows this triangle of numbers, when suitably normalized, satisfies the central limit theorem. - _Peter Bala_, Mar 05 2012
%C These are the coefficients of the midpoint Eulerian polynomials (see Quade/Collatz and Schoenberg). In terms of the cardinal B-splines b_n(t) these polynomials can be defined as M_n(x) = 2^n*n!*Sum_{k=0..n} b_{n+1}(k+1/2)*x^k. - _Peter Luschny_, Apr 26 2013
%C The o.g.f. Godd(n, x) = Sum_{m>=0} Sodd(n, m)*x^m, with Sodd(n, m) = Sum_{j=0..m} (1+2*j)^n is Podd(n, x)/(1 - x)^(n+2) with Podd(n, x) = Sum_{k=0..n} T(n+1, k+1)*x^k. E.g., Godd(2, x) = (1 + 6*x + x^2)/(1 - x)^4; see A000447(n+1) for n >= 0. For the e.g.f.s see A282628. - _Wolfdieter Lang_, Mar 17 2017
%C Let h_0(x,y) = x*y/(x+y), and D = x*D_x - y*D_y where D_x is the partial derivative w.r.t. x, etc. Put h_{n+1}(x,y) = D(h_n)(x,y). Then h_n(x,y) = x*y/(x+y)^(n+1)*f_{n}(x,y) where f_n(x,y) = Sum_{k=0..n} (-1)^k*T(n+1,k+1)*y^(n-k)*x^k. If instead of h_0, one similarly uses g_0(x,y) = x*y/(y-x), etc., then one obtains g_n(x,y) = x*y/(y-x)^(n+1)*Sum_{k=0..n} T(n+1,k+1)*y^(n-k)*x^k. (If instead of D one considers D' = x*D_x + y*D_y, then h_0 and g_0 are fixed points of D'.) - _Gregory Gerard Wojnar_, Oct 28 2018
%D G. Boros and V. H. Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, 2004.
%D T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Chapter 11.
%D W. Quade and L. Collatz, Zur Interpolationstheorie der reellen periodischen Funktionen. Sitzungsbericht der Preuss. Akad. der Wiss., Phys.-Math. Kl, (1938), 383-429.
%H Muniru A Asiru, <a href="/A060187/b060187.txt">Table of n, a(n) for n = 1..1275</a>
%H Jean-Christophe Aval, Adrien Boussicault, Philippe Nadeau, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i4p34">Tree-like Tableaux</a>, Electronic Journal of Combinatorics, 20(4), 2013, #P34.
%H Eli Bagno, David Garber, Mordechai Novick, <a href="https://arxiv.org/abs/2004.03681">The Worpitzky identity for the groups of signed and even-signed permutations</a>, arXiv:2004.03681 [math.CO], 2020.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry4/barry271.html">General Eulerian Polynomials as Moments Using Exponential Riordan Arrays</a>, Journal of Integer Sequences, 16 (2013), #13.9.6.
%H Paul Barry, <a href="https://arxiv.org/abs/1802.03443">On a transformation of Riordan moment sequences</a>, arXiv:1802.03443 [math.CO], 2018.
%H Paul Barry, <a href="https://arxiv.org/abs/1803.10297">Generalized Eulerian Triangles and Some Special Production Matrices</a>, arXiv:1803.10297 [math.CO], 2018.
%H Jose Bastidas, <a href="https://arxiv.org/abs/2009.05876">The polytope algebra of generalized permutahedra</a>, arXiv:2009.05876 [math.CO], 2020.
%H V. Batyrev and M. Blume, <a href="https://arxiv.org/abs/0912.2898">On generalizations of Losev-Manin moduli systems for classical root systems</a> arXiv:0912.2898 [math.AG], 2009-2011, (p. 13). - _Tom Copeland_, Oct 03 2014
%H Anna Borowiec, Wojciech Mlotkowski, <a href="https://arxiv.org/abs/1509.03758">New Eulerian numbers of type D</a>, arXiv:1509.03758 [math.CO], 2015.
%H Chak-On Chow and I. M. Gessel, <a href="http://people.brandeis.edu/~gessel/homepage/papers/chow.pdf">On the descent numbers and major indices for the hyperoctahedral group</a>, Adv. Appl. Math. 38, No. 3, 275-301 (2007).
%H Sandrine Dasse-Hartaut and Pawel Hitczenko, <a href="https://arxiv.org/abs/1202.3092">Greek letters in random staircase tableaux</a>, arXiv:1202.3092 [math.CO], 2012.
%H S. Fomin, N. Reading, <a href="https://arxiv.org/abs/math/0505518">Root systems and generalized associahedra</a>, Lecture notes for IAS/Park-City 2004, arXiv:math/0505518 [math.CO], 2005-2008.
%H Ghislain R. Franssens, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Franssens/franssens13.html">On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
%H P. Hitczenko and S. Janson, <a href="http://arxiv.org/abs/1212.5498">Weighted random staircase tableaux</a>, arXiv:1212.5498 [math.CO], 2012.
%H Svante Janson, <a href="https://arxiv.org/abs/1305.3512">Euler-Frobenius numbers and rounding</a>, arXiv:1305.3512 [math.PR], 2013.
%H Wolfdieter Lang, <a href="https://arxiv.org/abs/1707.04451">On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers</a>, arXiv:math/1707.04451 [math.NT], July 2017.
%H L. Liu, Y. Wang, <a href="https://arxiv.org/abs/math/0509207">A unified approach to polynomial sequences with only real zeros</a>, arXiv:math/0509207 [math.CO], 2005-2006.
%H Peter Luschny, <a href="http://www.luschny.de/math/euler/EulerianPolynomials.html">Eulerian polynomials.</a>
%H Shi-Mei Ma, <a href="https://arxiv.org/abs/1208.3104">Some combinatorial sequences associated with context-free grammars</a>, arXiv:1208.3104v2 [math.CO], 2012. - From N. J. A. Sloane, Aug 21 2012
%H Shi-Mei Ma, <a href="https://arxiv.org/abs/1304.6654">On gamma-vectors and the derivatives of the tangent and secant functions</a>, arXiv:1304.6654 [math.CO], 2013.
%H Shi-Mei Ma, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i1p11">A family of two-variable derivative polynomials for tangent and secant</a>, El J. Combinat. 20 (1) (2013) P11.
%H Shi-Mei Ma, Qi Fang, Toufik Mansour, Yeong-Nan Yeh, <a href="https://arxiv.org/abs/2104.09374">Alternating Eulerian polynomials and left peak polynomials</a>, arXiv:2104.09374, 2021
%H Shi-Mei Ma, Jun Ma, Yeong-Nan Yeh, <a href="https://arxiv.org/abs/1802.02861">On certain combinatorial expansions of descent polynomials and the change of grammars</a>, arXiv:1802.02861 [math.CO], 2018.
%H Shi-Mei Ma, T. Mansour, D. Callan, <a href="https://arxiv.org/abs/1404.0731">Some combinatorial arrays related to the Lotka-Volterra system</a>, arXiv:1404.0731 [math.CO], 2014.
%H Shi-Mei Ma, Hai-Na Wang, <a href="https://arxiv.org/abs/1506.08716">Enumeration of a dual set of Stirling permutations by their alternating runs</a>, arXiv:1506.08716 [math.CO], 2015.
%H P. A. MacMahon, <a href="http://dx.doi.org/10.1112/plms/s2-19.1.305">The divisors of numbers</a>, Proc. London Math. Soc., (2) 19 (1921), 305-340; Coll. Papers II, pp. 267-302.
%H F. Nakano, T. Sadahiro, <a href="https://arxiv.org/abs/1306.2790">A generalization of carries process and Eulerian numbers</a>, arXiv:1306.2790 [math.PR], 2013.
%H G. Rzadkowski, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Rzadkowski/rzad6.html">An Analytic Approach to Special Numbers and Polynomials</a>, J. Int. Seq. 18 (2015) 15.8.8.
%H I. J. Schoenberg, <a href="https://doi.org/10.1007/978-3-0348-7283-6_34">Cardinal interpolation and spline functions IV. The exponential Euler splines</a>. ISNM 20 (1972), 382-404.
%H R. P. Stanley and F. Zanello, <a href="https://arxiv.org/abs/1305.6083">Unimodality of partitions with distinct parts inside Ferrers shapes</a>, arXiv:1305.6083 [math.CO], 2013.
%H R. P. Stanley, F. Zanello, <a href="http://www-math.mit.edu/~rstan/papers/qbc.pdf">Some asymptotic results on q-binomial coefficients</a>, 2014.
%H Einar SteingrÃmsson, <a href="http://dx.doi.org/10.1006/eujc.1994.1021">Permutation statistics of indexed permutations</a>, European J. Combin. 15 (1994), no. 2, 187-205.
%H G. Strasser, <a href="http://dx.doi.org/10.1017/S0305004110000538">Generalisation of the Euler adic</a>, Math. Proc. Camb. Phil. Soc. 150 (2010) 241-256, Triangle A_2(n,k).
%F T(s, 2) = 3^(s-1) - s. Sum_{t=1..s} T(s, t) = 2^(s-1)*(s-1)!.
%F From _Peter Bala_, Oct 26 2008: (Start)
%F T(n,k) = Sum_{i = 1..k} (-1)^(k-i)*binomial(n,k-i)*(2*i-1)^(n-1).
%F E.g.f.: (1 - x)*exp((1 - x)*t)/(1 - x*exp(2*(1 - x)*t)) = 1 + (1 + x)*t + (1 + 6*x + x^2)*t^2/2! + ... .
%F The row polynomials R(n,x) satisfy R(n,x)/(1 - x)^n = Sum_{i >= 1} (2*i - 1)^(n-1)*x^i. For example, row 3 gives (x + 6*x^2 + x^3)/ (1 - x)^3 = x + 3^2*x^2 + 5 ^2*x^3 + 7^2*x^4 + ... .
%F The recurrence relation R(n+1,x) = [(2*n+1)*x - 1]*R(n,x) + 2*x*(1 - x)*R'(n,x) shows that the row polynomials R(n,x) have only real zeros (apply Corollary 1.2 of [Liu and Wang]).
%F Worpitzky-type identity: Sum_{k = 1..n} T(n,k)*binomial(x+k-1,n-1) = (2*x+1)^(n-1).
%F The nonzero alternating row sums are (-1)^(n-1)*A002436(n). (End)
%F exp(x)*(d/dx)^n [exp(x)/(1 - exp(2*x))] = R(n+1,exp(2*x))/ (1 - exp(2*x))^(n+1).
%F Compare with Example 12.3.1. in [Boros and Moll]. - _Peter Bala_, Nov 07 2008
%F The n-th row polynomial R(n,x) = Sum_{k = 0..n} A145901(n,k)*x^k*(1 - x)^(n-k) = Sum_{k = 0..n} A145901(n,k)*(x - 1)^(n-k). - _Peter Bala_, Jul 22 2014
%F Assuming an offset 0, the n-th row polynomial = (x - 1)^n * log(x) * Integral_{u = 0..inf} (2*floor(u) + 1)^n * x^(-u) du, provided x > 1. - _Peter Bala_, Feb 06 2015
%F The finite sums of consecutive odd integer powers is derived from this number triangle: Sum_{k=1..n}(2k-1)^m = Sum_{j=1..m+1}binomial(n+m+1-j,m+1)*T(m+1,j). - _Tony Foster III_, Feb 09 2018
%e The triangle T(n, k) begins:
%e n\k 1 2 3 4 5 6 7 8 ...
%e 1: 1
%e 2: 1 1
%e 3: 1 6 1
%e 4: 1 23 23 1
%e 5: 1 76 230 76 1
%e 6: 1 237 1682 1682 237 1
%e 7: 1 722 10543 23548 10543 722 1
%e 8: 1 2179 60657 259723 259723 60657 2179 1
%e ...
%e row n = 9: 1 6552 331612 2485288 4675014 2485288 331612 6552 1,
%e row n = 10: 1 19673 1756340 21707972 69413294 69413294 21707972 1756340 19673 1,
%e row n = 11: 1 59038 9116141 178300904 906923282 1527092468 906923282 178300904 9116141 59038 1, ... reformatted. - _Wolfdieter Lang_, Mar 17 2017
%p A060187:= (n,k) -> add((-1)^(k-i)*binomial(n,k-i)*(2*i-1)^(n-1), i = 1..k):
%p for n from 1 to 10 do seq(A060187(n,k),k = 1..n); end do; # _Peter Bala_, Oct 26 2008
%p T:=proc(n,k,l) option remember; if (n=1 or k=1 or k=n) then 1 else
%p (l*n-l*k+1)*T(n-1,k-1,l)+(l*k-l+1)*T(n-1,k,l); fi; end;
%p for n from 1 to 10 do lprint([seq(T(n,k,2),k=1..n)]); od; # _N. J. A. Sloane_, May 08 2013
%p P := proc(n,x) option remember; if n = 0 then 1 else
%p (n*x+(1/2)*(1-x))*P(n-1,x)+x*(1-x)*diff(P(n-1,x),x);
%p expand(%) fi end:
%p A060187 := (n,k) -> 2^n*coeff(P(n,x),x,k):
%p seq(print(seq(A060187(n,k), k=0..n)), n=0..10); # _Peter Luschny_, Mar 08 2014
%t p[x_, n_] = 2^n (1 - x)^(1 + n) LerchPhi[x, -n, 1/2]; Table[CoefficientList[p[x, n], x], {n, 0, 10}] // Flatten (* _Roger L. Bagula_, Sep 16 2008 *)
%t T[n_, k_] := Sum[(-1)^(k-i)*Binomial[n, k-i]*(2*i-1)^(n-1), {i, 1, k}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Nov 23 2015, after _Peter Bala_ *)
%o (PARI) {T(n, k) = if( n<k || k<1, 0, sum(i=1, k, (-1)^(k-i) * binomial(n, k-i) * (2*i-1)^(n-1)))}; /* _Michael Somos_, Jan 07 2011 */
%o (Sage)
%o @CachedFunction
%o def A060187(n, k) :
%o if n == 0: return 1 if k == 0 else 0
%o return (2*(n-k)+1)*A060187(n-1, k-1) + (2*k+1)*A060187(n-1, k)
%o for n in (0..8): [A060187(n,k) for k in (0..n)] # _Peter Luschny_, Apr 26 2013
%o (GAP) a:=Flat(List([1..11],n->List([1..n],k->Sum([1..k],i->(-1)^(k-i)*Binomial(n,k-i)*(2*i-1)^(n-1))))); # _Muniru A Asiru_, Feb 09 2018
%o (Magma) [[(&+[(-1)^(k-j)*Binomial(n,k-j)*(2*j-1)^(n-1): j in [1..k]]): k in [1..n]]: n in [1..10]]; // _G. C. Greubel_, Nov 08 2018
%Y Diagonals give A060188, A060189, A060190. Cf. A008292.
%Y Cf. also A000165 (row sums), A002436 (alt. row sums), A008292, A145901, A145905 (Hilbert transform). A062715.
%K nonn,tabl,easy,nice
%O 1,5
%A _N. J. A. Sloane_, Mar 20 2001
|
