%I #44 Jun 03 2022 04:54:21
%S 1,1,1,1,5,1,1,14,21,1,1,30,147,85,1,1,55,627,1408,341,1,1,91,2002,
%T 11440,13013,1365,1,1,140,5278,61490,196053,118482,5461,1,1,204,12138,
%U 251498,1733303,3255330,1071799,21845,1,1,285,25194,846260,10787231
%N Triangle of central factorial numbers T(2*n,2*n-2*k), k >= 0, n >= 1 (in Riordan's notation).
%C D. E. Knuth [1992] on page 10 gives the formula Sum n^(2m-1) = Sum_{k=1..m} (2k-1)! T(2m,2k) binomial(n+k, 2k) where T(m, k) is the central factorial number of the second kind. - _Michael Somos_, May 08 2018
%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217, Table 6.2(a).
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.8.
%H Reinhard Zumkeller, <a href="/A008957/b008957.txt">Rows n = 1..100 of triangle, flattened</a>
%H F. Alayont and N. Krzywonos, <a href="http://faculty.gvsu.edu/alayontf/notes/rook_polynomials_higher_dimensions_preprint.pdf">Rook Polynomials in Three and Higher Dimensions</a>, 2012. [From _N. J. A. Sloane_, Jan 02 2013]
%H D. Dumont, <a href="http://dx.doi.org/10.1215/S0012-7094-74-04134-9">Interprétations combinatoires des nombres de Genocchi</a>, Duke Math. J., 41 (1974), 305-318.
%H D. E. Knuth, <a href="https://arxiv.org/abs/math/9207222">Johann Faulhaber and Sums of Powers</a>, arXiv:math/9207222, Jul 1992. See bottom of page 10. [From _Michael Somos_, May 08 2018]
%F From _Michael Somos_, May 08 2018: (Start)
%F T(n, k) = T(n-1, k-1) + k^2 * T(n-1, k), where T(n, n) = T(n, 1) = 1.
%F E.g.f.: x^2 * cosh(sinh(y*x/2) / (x/2)) - 1) = (1*x^2)*y^2/2! + (1*x^2 + 1*x^4)*y^4/4! +(1*x^2 + 5*x^4 + x^6)*y^6/6! + (1*x^2 + 14*x^4 + 21*x^6 + 1*x^8)*y^8/8! + ... (End)
%e The triangle starts:
%e 1;
%e 1, 1;
%e 1, 5, 1;
%e 1, 14, 21, 1;
%e 1, 30, 147, 85, 1;
%e 1, 55, 627, 1408, 341, 1;
%e 1, 91, 2002, 11440, 13013, 1365, 1;
%p A036969 := proc(n,k) local j; 2*add(j^(2*n)*(-1)^(k-j)/((k-j)!*(k+j)!),j=1..k); end; # Gives rows of triangle in reversed order
%t t[n_, n_] = t[n_, 1] = 1;
%t t[n_, k_] := t[n-1, k-1] + k^2 t[n-1, k];
%t Flatten[Table[t[n, k], {n, 1, 10}, {k, n, 1, -1}]][[1 ;; 50]] (* _Jean-François Alcover_, Jun 16 2011 *)
%o (Haskell)
%o a008957 n k = a008957_tabl !! (n-1) (k-1)
%o a008957_row n = a008957_tabl !! (n-1)
%o a008957_tabl = map reverse a036969_tabl
%o -- _Reinhard Zumkeller_, Feb 18 2013
%o (PARI) {T(n, k) = if( n<1 || k>n, 0, n==k || k==1, 1, T(n-1, k-1) + k^2 * T(n-1, k))}; \\ _Michael Somos_, May 08 2018
%o (Sage)
%o def A008957(n, k):
%o m = n - k
%o return 2*sum((-1)^(j+m)*(j+1)^(2*n)/(factorial(j+m+2)*factorial(m-j)) for j in (0..m))
%o for n in (1..7): print([A008957(n, k) for k in (1..n)]) # _Peter Luschny_, May 10 2018
%Y Row reversed version of A036969. (0,0)-based version: A269945.
%Y Cf. A008955.
%K nonn,nice,easy,tabl
%O 1,5
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Apr 16 2000