A000497 S2(j,2j+2) where S2(n,k) is a 2-associated Stirling number of the second kind.

1, 25, 490, 9450, 190575, 4099095, 94594500, 2343240900, 62199262125, 1764494857125, 53338158823950, 1712934942468750, 58274046742786875, 2094379201311271875, 79318164037837725000, 3157886388887074845000

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
• F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 296.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..100
• H. W. Gould, Harris Kwong, Jocelyn Quaintance, On Certain Sums of Stirling Numbers with Binomial Coefficients, J. Integer Sequences, 18 (2015), #15.9.6.
• M. Ward, The representations of Stirling's numbers and Stirling's polynomials as sums of factorials, Amer. J. Math., 56 (1934), p. 87-95.

Crossrefs

Cf. A008299, A000504.

Programs

Maple

gf := (u,t)->exp(u*(exp(t)-1-t)); S2a := j->simplify(subs(u=0,t=0,diff(gf(u,t),u$j,t$(2*j+2)))/j!); for i from 1 to 20 do S2a(i); od;
# Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 12 2000

Mathematica

t[n_, k_] := Sum[ (-1)^i*Binomial[n, i]*Sum[ (-1)^j*(k-i-j)^(n-i)/(j!*(k-i-j)!), {j, 0, k-i}], {i, 0, k}]; Table[ t[2n+2, n], {n, 1, 16}  ](* Jean-François Alcover, Feb 24 2012 *)
Table[n*(n+1)*(2*n+1)*2^n*Gamma[n+3/2]/(9*Sqrt[Pi]),{n,1,20}] (* Vaclav Kotesovec, Aug 07 2013 *)

Formula

G.f.: x*(4*x+1)*hypergeom([3, 7/2],[],2*x)+28*x^3*hypergeom([4, 9/2],[],2*x). - Mark van Hoeij, Apr 07 2013

a(n) = n*(n+1)*(2*n+1)*2^n*GAMMA(n+3/2)/(9*sqrt(Pi)). - Vaclav Kotesovec, Aug 07 2013

(2*n-1)*(n-1)*a(n) -(n+1)*(1+2*n)^2*a(n-1)=0. - R. J. Mathar, Jun 09 2018

Extensions

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 12 2000