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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). 6
 1, 1, 1, 1, 5, 1, 1, 14, 21, 1, 1, 30, 147, 85, 1, 1, 55, 627, 1408, 341, 1, 1, 91, 2002, 11440, 13013, 1365, 1, 1, 140, 5278, 61490, 196053, 118482, 5461, 1, 1, 204, 12138, 251498, 1733303, 3255330, 1071799, 21845, 1, 1, 285, 25194, 846260, 10787231 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS 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 REFERENCES J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217, Table 6.2(a). R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.8. LINKS Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened F. Alayont and N. Krzywonos, Rook Polynomials in Three and Higher Dimensions, 2012. [From N. J. A. Sloane, Jan 02 2013] D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318. D. E. Knuth, Johann Faulhaber and Sums of Powers, arXiv:math/9207222, Jul 1992. See bottom of page 10. [From Michael Somos, May 08 2018] FORMULA From Michael Somos, May 08 2018: (Start) T(n, k) = T(n-1, k-1) + k^2 * T(n-1, k), where T(n, n) = T(n, 1) = 1. 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) EXAMPLE The triangle starts: 1; 1,  1; 1,  5,    1; 1, 14,   21,     1; 1, 30,  147,    85,     1; 1, 55,  627,  1408,   341,    1; 1, 91, 2002, 11440, 13013, 1365, 1; MAPLE 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 MATHEMATICA t[n_, n_] = t[n_, 1] = 1; t[n_, k_] := t[n-1, k-1] + k^2 t[n-1, k]; Flatten[Table[t[n, k], {n, 1, 10}, {k, n, 1, -1}]][[1 ;; 50]] (* Jean-François Alcover, Jun 16 2011 *) PROG (Haskell) a008957 n k = a008957_tabl !! (n-1) (k-1) a008957_row n = a008957_tabl !! (n-1) a008957_tabl = map reverse a036969_tabl -- Reinhard Zumkeller, Feb 18 2013 (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 (Sage) def A008957(n, k):     m = n - k     return 2*sum((-1)^(j+m)*(j+1)^(2*n)/(factorial(j+m+2)*factorial(m-j)) for j in (0..m)) for n in (1..7): print([A008957(n, k) for k in (1..n)]) # Peter Luschny, May 10 2018 CROSSREFS Row reversed version of A036969. Cf. A008955. Sequence in context: A181143 A144438 A157207 * A136267 A109960 A196019 Adjacent sequences:  A008954 A008955 A008956 * A008958 A008959 A008960 KEYWORD nonn,nice,easy,tabl AUTHOR EXTENSIONS More terms from Vladeta Jovovic, Apr 16 2000 STATUS approved

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Last modified September 20 11:01 EDT 2020. Contains 337264 sequences. (Running on oeis4.)