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%I #69 Mar 11 2024 08:29:21
%S 1,1,2,1,4,3,1,6,9,4,1,8,19,16,5,1,10,33,44,25,6,1,12,51,96,85,36,7,1,
%T 14,73,180,225,146,49,8,1,16,99,304,501,456,231,64,9,1,18,129,476,985,
%U 1182,833,344,81,10
%N Table of figurate numbers for the n-dimensional cross polytopes.
%C The n-th row entries for this array are the regular polytope numbers for the n-dimensional cross polytope as defined by [Kim]. The rows are the partial sums of the rows of the square array of Delannoy numbers A008288.
%C The odd numbered rows of this array form A142977. For a triangular version of this table see A104698. Cf. also A101603.
%C The n-th row of the array is the binomial transform of n-th row of triangle A081277, followed by zeros. Example: row 4 (1, 6, 19, 44, 85, ...) = binomial transform of row 3 of A081277: (1, 5, 8, 4, 0, 0, 0, ...). - _Gary W. Adamson_, Jul 17 2008
%C The main diagonal of the array T(n,k) is A047781 Sum_{k=0..n-1} binomial(n-1,k)*binomial(n+k,k). Also a(n) = T(n,n), array T as in A049600. The link from A099193 to J. V. Post, Table of polytope numbers, Sorted, Through 1,000,000, includes all n-D Hyperoctahedron (n-Cross Polytope) Numbers through 10-Cross(20) = 1669752016. - _Jonathan Vos Post_, Jul 16 2008
%H Reinhard Zumkeller, <a href="/A142978/b142978.txt">Rows n = 1..100 of triangle, flattened</a>
%H Peter Bala, <a href="/A142978/a142978.pdf">A142978 and a family of continued fraction expansions for log(2)</a>
%H Valentin Bonzom and Etera R. Livine, <a href="https://arxiv.org/abs/1905.00348">Self-duality of the 6j-symbol and Fisher zeros for the Tetrahedron</a>, arXiv:1905.00348 [math-ph], 2019.
%H Steven Edwards and William Griffiths, <a href="https://www.fq.math.ca/Papers1/55-4/EdwardsGriffiths82617.pdf">Generalizations of Delannoy and cross polytope numbers</a>, Fib. Q., Vol. 55, No. 4 (2017), pp. 356-366.
%H Steven Edwards and William Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Griffiths/griffiths51.html">On Generalized Delannoy Numbers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
%H Milan Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.
%H Milan Janjic and B. Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014) # 14.3.5.
%H Hyun Kwang Kim, <a href="http://dx.doi.org/10.1090/S0002-9939-02-06710-2">On Regular Polytope Numbers</a>, Proc. Amer. Math. Soc., 131 (2002), 65-75.
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/MeixnerPolynomialoftheFirstKind.html">Meixner polynomial of the first kind</a>
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/Mittag-LefflerPolynomial.html">Mittag-Leffler polynomial</a>
%H Yutong Yang, <a href="http://digitalcommons.kennesaw.edu/honors_etd/13">From Simplest Recursion to the Recursion of Generalizations of Cross Polytope Numbers</a>, (2017), Honors College Capstone and Theses, 13.
%F T(n,k) = Sum_{i = 0..n-1} C(n-1,i)*C(k+i,n).
%F Reciprocity law: n*T(n,k) = k*T(k,n).
%F Recurrence relation: T(n,1) = 1, T(1,k) = k, T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k), n,k > 1.
%F O.g.f. row n: x*(1 + x)^(n-1)/(1 - x)^(n+1).
%F O.g.f. for array: Sum_{n >= 1, k >= 1} T(n, k)*x^k*y^n = x*y/((1 - x)*(1 - x - y - x*y)).
%F The n-th row entries are the values [p_n(k)], k >= 1, of the polynomial function p_n(x) = Sum_{k = 1..n} 2^(k-1)*C(n-1,k-1)*C(x,k). The first few values are p_1(x) = x, p_2(x) = x^2, p_3(x) = (2*x^3 + x)/3 and p_4(x) = (x^4 + 2*x^2)/3.
%F The polynomial p_n(x) is the unique polynomial solution of the difference equation x*( f(x+1) - f(x-1) ) = 2*n*f(x), normalized so that f(1) = 1.
%F The o.g.f. for the p_n(x) is 1/2*((1 + t)/(1 - t))^x = 1/2 + x*t + x^2*t^2 + (2*x^3 + x)/3*t^3 + .... Thus p_n(x) is, apart from a constant factor, the Meixner polynomial of the first kind M_n(x;b,c) at b = 0, c = -1, also known as a Mittag-Leffler polynomial.
%F The entries in the n-th row appear in the series acceleration formula for the constant log(2): Sum_{k >= 1} (-1)^(k+1)/(T(n,k)*T(n,k+1)) = 1 + (-1)^(n+1) * (2*n)*(log(2) - (1 - 1/2 + 1/3 - ... + (-1)^(n+1)/n)). For example, n = 3 gives log(2) = 4/6 + (1/6)*(1/(1*6) - 1/(6*19) + 1/(19*44) - 1/(44*85) + ...). See A142983 for further details.
%F From _Peter Bala_, Oct 02 2008: (Start)
%F The odd-indexed columns of this array form the array A142992 of crystal ball sequences for lattices of type C_n.
%F Conjectural congruences for main diagonal entries: Put A(n) = T(n,n). Calculation suggests the following congruences: for prime p > 3 and m, r >= 1, A(m*p^r) == A(m*p^(r-1)) (mod p^(3*r));
%F Sum_{k = 0..p-1} A(k)^2 == 0 (mod p) if p is a prime of the form 8*n+1 or 8*n+7;
%F Sum_{k = 0..p-1} A(k)^2 == -1 (mod p) if p is a prime of the form 8*n+3 or 8*n+5.
%F (End)
%F From _Peter Bala_, Sep 27 2021: (Start)
%F T(n,k) = (1/2)*Sum_{i = 0..k} binomial(k,i)*binomial(n+k-1-i,k-1).
%F T(n,k) = (1/2)*[x^n] ((1+x)/(1-x))^k = (1/2)*(k/n)*[x^k] ((1+x)/(1-x))^n.
%F n*T(n,k) = 2*k*T(n-1,k) + (n - 2)*T(n-2,k). (End)
%F A(n,k) = k*hypergeom([1 - n, 1 - k], [2], 2). - _Peter Luschny_, Mar 23 2023
%e The square array A(n, k) begins:
%e n\k| 1 2 3 4 5 6
%e ---+-------------------------------
%e 1 | 1 2 3 4 5 6 A000027
%e 2 | 1 4 9 16 25 36 A000290
%e 3 | 1 6 19 44 85 146 A005900
%e 4 | 1 8 33 96 225 456 A014820
%e 5 | 1 10 51 180 501 1182 A069038
%e 6 | 1 12 73 304 985 2668 A069039
%e 7 | 1 14 99 476 1765 5418 A099193
%p with(combinat): T:=(n,k) -> add(binomial(n-1,i)*binomial(k+i,n),i = 0..n-1); for n from 1 to 10 do seq(T(n,k),k = 1..10) end do; # Program restored by _Peter Bala_, Oct 02 2008
%p A := (n, k) -> k*hypergeom([1 - n, 1 - k], [2], 2):
%p seq(print(seq(simplify(A(n, k)), k = 1..9)), n=1..7); # _Peter Luschny_, Mar 23 2023
%t t[n_, k_] := Sum[ Binomial[n-1, i]*Binomial[k+i, n], {i, 0, n-1}]; Table[t[n-k, k], {n, 1, 11}, {k, 1, n-1}] // Flatten (* _Jean-François Alcover_, Mar 06 2013 *)
%o (Haskell)
%o a142978 n k = a142978_tabl !! (n-1) !! (k-1)
%o a142978_row n = a142978_tabl !! (n-1)
%o a142978_tabl = map reverse a104698_tabl
%o -- _Reinhard Zumkeller_, Jul 17 2015
%Y Cf. A008288 (Delannoy numbers), A005900 (row 3), A014820 (row 4), A069038 (row 5), A069039 (row 6), A099193 (row 7), A099195 (row 8), A099196 (row 9), A099197 (row 10), A101603, A104698 (triangle version), A142977, A142983.
%Y Cf. A049600, A047781, A081277.
%K easy,nonn,tabl
%O 1,3
%A _Peter Bala_, Jul 15 2008
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