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A008957 Triangle of central factorial numbers T(2*n,2*n-2*k), k >= 0, n >= 1 (in Riordan's notation).

%I A008957 #52 Dec 29 2024 20:25:07
%S A008957 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 A008957 11440,13013,1365,1,1,140,5278,61490,196053,118482,5461,1,1,204,12138,
%U A008957 251498,1733303,3255330,1071799,21845,1,1,285,25194,846260,10787231
%N A008957 Triangle of central factorial numbers T(2*n,2*n-2*k), k >= 0, n >= 1 (in Riordan's notation).
%C A008957 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 A008957 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217, Table 6.2(a).
%D A008957 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.8.
%H A008957 Reinhard Zumkeller, <a href="/A008957/b008957.txt">Rows n = 1..100 of triangle, flattened</a>
%H A008957 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 A008957 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 A008957 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]
%H A008957 Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/PolynomialIdentitiesInvolvingCentralFactorialNumbers.pdf">Polynomial identities involving central factorial numbers</a>, GitHub, 2024. See p. 6.
%F A008957 From _Michael Somos_, May 08 2018: (Start)
%F A008957 T(n, k) = T(n-1, k-1) + k^2 * T(n-1, k), where T(n, n) = T(n, 1) = 1.
%F A008957 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 + 1*x^6)*y^6/6! + (1*x^2 + 14*x^4 + 21*x^6 + 1*x^8)*y^8/8! + ... (End)
%e A008957 The triangle starts:
%e A008957   1;
%e A008957   1,  1;
%e A008957   1,  5,    1;
%e A008957   1, 14,   21,     1;
%e A008957   1, 30,  147,    85,     1;
%e A008957   1, 55,  627,  1408,   341,    1;
%e A008957   1, 91, 2002, 11440, 13013, 1365, 1;
%p A008957 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 A008957 t[n_, n_] = t[n_, 1] = 1;
%t A008957 t[n_, k_] := t[n-1, k-1] + k^2 t[n-1, k];
%t A008957 Flatten[Table[t[n, k], {n, 1, 10}, {k, n, 1, -1}]][[1 ;; 50]] (* _Jean-François Alcover_, Jun 16 2011 *)
%o A008957 (Haskell)
%o A008957 a008957 n k = a008957_tabl !! (n-1) (k-1)
%o A008957 a008957_row n = a008957_tabl !! (n-1)
%o A008957 a008957_tabl = map reverse a036969_tabl
%o A008957 -- _Reinhard Zumkeller_, Feb 18 2013
%o A008957 (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 A008957 (Sage)
%o A008957 def A008957(n, k):
%o A008957     m = n - k
%o A008957     return 2*sum((-1)^(j+m)*(j+1)^(2*n)/(factorial(j+m+2)*factorial(m-j)) for j in (0..m))
%o A008957 for n in (1..7): print([A008957(n, k) for k in (1..n)]) # _Peter Luschny_, May 10 2018
%Y A008957 Row reversed version of A036969. (0,0)-based version: A269945.
%Y A008957 Cf. A008955.
%K A008957 nonn,nice,easy,tabl
%O A008957 1,5
%A A008957 _N. J. A. Sloane_
%E A008957 More terms from _Vladeta Jovovic_, Apr 16 2000