|
%I #182 Nov 30 2023 23:40:13
%S 1,1,1,3,3,1,15,15,6,1,105,105,45,10,1,945,945,420,105,15,1,10395,
%T 10395,4725,1260,210,21,1,135135,135135,62370,17325,3150,378,28,1,
%U 2027025,2027025,945945,270270,51975,6930,630,36,1,34459425,34459425,16216200,4729725,945945,135135,13860,990,45,1
%N Triangle of coefficients of Bessel polynomials (exponents in decreasing order).
%C The (reverse) Bessel polynomials P(n,x):=Sum_{m=0..n} a(n,m)*x^m, the row polynomials, called Theta_n(x) in the Grosswald reference, solve x*(d^2/dx^2)P(n,x) - 2*(x+n)*(d/dx)P(n,x) + 2*n*P(n,x) = 0.
%C With the related Sheffer associated polynomials defined by Carlitz as
%C B(0,x) = 1
%C B(1,x) = x
%C B(2,x) = x + x^2
%C B(3,x) = 3 x + 3 x^2 + x^3
%C B(4,x) = 15 x + 15 x^2 + 6 x^3 + x^4
%C ... (see Mathworld reference), then P(n,x) = 2^n * B(n,x/2) are the Sheffer polynomials described in A119274. - _Tom Copeland_, Feb 10 2008
%C Exponential Riordan array [1/sqrt(1-2x), 1-sqrt(1-2x)]. - _Paul Barry_, Jul 27 2010
%C From _Vladimir Kruchinin_, Mar 18 2011: (Start)
%C For B(n,k){...} the Bell polynomial of the second kind we have
%C B(n,k){f', f'', f''', ...} = T(n-1,k-1)*(1-2*x)^(k/2-n), where f(x) = (1-sqrt(1-2*x).
%C The expansions of the first few rows are:
%C 1/sqrt(1-2*x);
%C 1/(1-2*x)^(3/2), 1/(1-2*x);
%C 3/(1-2*x)^(5/2), 3/(1-2*x)^2, 1/(1-2*x)^(3/2);
%C 15/(1-2*x)^(7/2), 15/(1-2*x)^3, 6/(1-2*x)^(5/2), 1/(1-2*x)^2. (End)
%C Also the Bell transform of A001147 (whithout column 0 which is 1,0,0,...). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 19 2016
%C Antidiagonals of A099174 are rows of this entry. Dividing each diagonal by its first element generates A054142. - _Tom Copeland_, Oct 04 2016
%C The row polynomials p_n(x) of A107102 are (-1)^n B_n(1-x), where B_n(x) are the modified Carlitz-Bessel polynomials above, e.g., (-1)^2 B_2(1-x) = (1-x) + (1-x)^2 = 2 - 3 x + x^2 = p_2(x). - _Tom Copeland_, Oct 10 2016
%C a(n-1,m-1) counts rooted unordered binary forests with n labeled leaves and m roots. - _David desJardins_, Feb 23 2019
%C From _Jianing Song_, Nov 29 2021: (Start)
%C The polynomials P_n(x) = Sum_{k=0..n} T(n,k)*x^k satisfy: P_n(x) - (d/dx)P_n(x) = x*P_{n-1}(x) for n >= 1.
%C {P(n,x)} are related to the Fourier transform of 1/(1+x^2)^(n+1) and x/(1+x^2)^(n+2):
%C (i) For n >= 0, real number t, we have Integral_{x=-oo..oo} exp(-i*t*x)/(1+x^2)^(n+1) dx = Pi/(2^n*n!) * P_n(|t|) * exp(-|t|);
%C (ii) For n >= 0, real number t, we have Integral_{x=-oo..oo} x*exp(-i*t*x)/(1+x^2)^(n+2) dx = Pi/(2^(n+1)*(n+1)!) * ((-t)*P_n(-|t|)) * exp(-|t|). (End)
%C Suppose that f(x) is an n-times differentiable function defined on (a,b) for 0 <= a < b <= +oo, then for n >= 1, the n-th derivative of f(sqrt(x)) on (a^2,b^2) is Sum_{k=1..n} ((-1)^(n-k)*T(n-1,k-1)*f^(k)(sqrt(x))) / (2^n*x^(n-(k/2))), where f^(k) is the k-th derivative of f. - _Jianing Song_, Nov 30 2023
%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
%H T. D. Noe, <a href="/A001497/b001497.txt">Rows n = 0..50 of triangle, flattened</a>
%H Peter Bala, <a href="/A035342/a035342_Bala.txt">Generalized Dobinski formulas</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry1/barry97r2.html">Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms</a>, J. Int. Seq. 14 (2011) # 11.2.2, chapter 8.
%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/08/23/a-class-of-differential-operators-and-the-stirling-numbers/">A Class of Differential Operators and the Stirling Numbers</a>
%H E. Deutsch, L. Ferrari and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.aam.2004.05.002">Production Matrices</a>, Advances in Applied Mathematics, 34 (2005) pp. 101-122.
%H O. Frink and H. L. Krall, <a href="http://dx.doi.org/10.1090/S0002-9947-1949-0028473-1">A new class of orthogonal polynomials</a>, Trans. Amer. Math. Soc. 65,100-115, 1945. [From _Roger L. Bagula_, Feb 15 2009]
%H E. Grosswald, <a href="http://dx.doi.org/10.1007/BFb0063135">Bessel Polynomials</a>, Lecture Notes Math. vol. 698 1978 p. 18.
%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Janjic/janjic22.html">Some classes of numbers and derivatives</a>, JIS 12 (2009) #09.8.3.
%H Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H B. Leclerc, <a href="http://dx.doi.org/10.1016/0012-365X(95)00138-M">Powers of staircase Schur functions and symmetric analogues of Bessel polynomials</a>, Discrete Math., 153 (1996), 213-227.
%H Robert S. Maier, <a href="https://arxiv.org/abs/2308.10332">Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers</a>, arXiv:2308.10332 [math.CO], 2023. See p. 19.
%H Toufik Mansour, Matthias Schork and Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Schork/schork2.html">The Generalized Stirling and Bell Numbers Revisited</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
%H Toufik Mansour, Matthias Schork and Mark Shattuck, <a href="http://dx.doi.org/10.1016/j.aml.2012.02.009">On the Stirling numbers associated with the meromorphic Weyl algebra</a>, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771. - From _N. J. A. Sloane_, Sep 15 2012
%H W. Mlotkowski and A. Romanowicz, <a href="http://www.math.uni.wroc.pl/~pms/files/33.2/Article/33.2.19.pdf">A family of sequences of binomial type</a>, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
%H Mathias Pétréolle and Alan D. Sokal, <a href="https://arxiv.org/abs/1907.02645">Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions</a>, arXiv:1907.02645 [math.CO], 2019.
%H Feng Qi and Bai-Ni Guo, <a href="https://doi.org/10.1002/9781119414421.ch5">"Some Properties and Generalizations of the Catalan, Fuss, and Fuss-Catalan Numbers"</a>, Mathematical Analysis and Applications : Selected Topics (2018), Wiley, Ch. 5, 101-133.
%H Feng Qi, X.-T. Shi and F.-F. Liu, <a href="https://www.researchgate.net/publication/280884520">Several formulas for special values of the Bell polynomials of the second kind and applications</a>, Preprint 2015.
%H Alexander Stoimenow, <a href="https://doi.org/10.1016/S0012-365X(99)00347-7">On the number of chord diagrams</a>, Discr. Math. 218 (2000), 209-233. Lemma 2.2.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BesselPolynomial.html">Bessel Polynomial</a>
%H <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>
%F a(n, m) = (2*n-m)!/(m!*(n-m)!*2^(n-m)) if n >= m >= 0 else 0 (from Grosswald, p. 7).
%F a(n, m)= 0, n<m; a(n, -1) := 0; a(0, 0)= 1; a(n, m) = (2*n-m-1)*a(n-1, m) + a(n-1, m-1), n >= m >= 0 (from Grosswald p. 23, (19)).
%F E.g.f. for m-th column: ((1-sqrt(1-2*x))^m)/(m!*sqrt(1-2*x)).
%F G.f.: 1/(1-xy-x/(1-xy-2x/(1-xy-3x/(1-xy-4x/(1-.... (continued fraction). - _Paul Barry_, Jan 29 2009
%F T(n,k) = if(k<=n, C(2n-k,2(n-k))*(2(n-k)-1)!!,0) = if(k<=n, C(2n-k,2(n-k))*A001147(n-k),0). - _Paul Barry_, Mar 18 2011
%F Row polynomials for n>=1 are given by 1/t*D^n(exp(x*t)) evaluated at x = 0, where D is the operator 1/(1-x)*d/dx. - _Peter Bala_, Nov 25 2011
%F The matrix product A039683*A008277 gives a signed version of this triangle. Dobinski-type formula for the row polynomials: R(n,x) = (-1)^n*exp(x)*Sum_{k = 0..inf} k*(k-2)*(k-4)*...*(k-2*(n-1))*(-x)^k/k!. Cf. A122850. - _Peter Bala_, Jun 23 2014
%e Triangle begins
%e 1,
%e 1, 1,
%e 3, 3, 1,
%e 15, 15, 6, 1,
%e 105, 105, 45, 10, 1,
%e 945, 945, 420, 105, 15, 1,
%e 10395, 10395, 4725, 1260, 210, 21, 1,
%e 135135, 135135, 62370, 17325, 3150, 378, 28, 1,
%e 2027025, 2027025, 945945, 270270, 51975, 6930, 630, 36, 1
%e Production matrix begins
%e 1, 1,
%e 2, 2, 1,
%e 6, 6, 3, 1,
%e 24, 24, 12, 4, 1,
%e 120, 120, 60, 20, 5, 1,
%e 720, 720, 360, 120, 30, 6, 1,
%e 5040, 5040, 2520, 840, 210, 42, 7, 1,
%e 40320, 40320, 20160, 6720, 1680, 336, 56, 8, 1,
%e 362880, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1
%e This is the exponential Riordan array A094587, or [1/(1-x),x], beheaded.
%e - _Paul Barry_, Mar 18 2011
%p f := proc(n) option remember; if n <=1 then (1+x)^n else expand((2*n-1)*x*f(n-1)+f(n-2)); fi; end;
%p row := n -> seq(coeff(f(n), x, n - k), k = 0..n): seq(row(n), n = 0..9);
%t m = 9; Flatten[ Table[(n + k)!/(2^k*k!*(n - k)!), {n, 0, m}, {k, n, 0, -1}]] (* _Jean-François Alcover_, Sep 20 2011 *)
%t y[n_, x_] := Sqrt[2/(Pi*x)]*E^(1/x)*BesselK[-n-1/2, 1/x]; t[n_, k_] := Coefficient[y[n, x], x, k]; Table[t[n, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Mar 01 2013 *)
%o (PARI) T(k, n) = if(n>k||k<0||n<0,0,(2*k-n)!/(n!*(k-n)!*2^(k-n))) /* _Ralf Stephan_ */
%o (PARI) {T(n, k) = if( k<0 || k>n, 0, binomial(n, k)*(2*n-k)!/2^(n-k)/n!)}; /* _Michael Somos_, Oct 03 2006 */
%o (Haskell)
%o a001497 n k = a001497_tabl !! n !! k
%o a001497_row n = a001497_tabl !! n
%o a001497_tabl = [1] : f [1] 1 where
%o f xs z = ys : f ys (z + 2) where
%o ys = zipWith (+) ([0] ++ xs) (zipWith (*) [z, z-1 ..] (xs ++ [0]))
%o -- _Reinhard Zumkeller_, Jul 11 2014
%o (Magma) /* As triangle */ [[Factorial(2*n-k)/(Factorial(k)*Factorial(n-k)*2^(n-k)): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Aug 12 2015
%o (Sage) # uses[bell_matrix from A264428]
%o # Adds a column 1,0,0,0, ... at the left side of the triangle.
%o bell_matrix(lambda n: A001147(n), 9) # _Peter Luschny_, Jan 19 2016
%Y Reflected version of A001498 which is considered the main entry.
%Y Other versions of this same triangle are given in A144299, A111924 and A100861.
%Y Row sums give A001515. a(n, 0)= A001147(n) (double factorials).
%Y Cf. A104556 (matrix inverse). A039683, A122850.
%Y Cf. A245066 (central terms).
%Y Cf. A054142, A099174, A107102.
%K nonn,tabl,nice
%O 0,4
%A _N. J. A. Sloane_
|
