A000504 S2(j,2j+3) where S2(n,k) is a 2-associated Stirling number of the second kind.

1, 56, 1918, 56980, 1636635, 47507460, 1422280860, 44346982680, 1446733012725, 49473074851200, 1774073543492250, 66681131440423500, 2624634287988087375, 108060337458000427500, 4647703259223579555000, 208548093035794902390000, 9749651260035434678555625

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
• 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).
• F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 296.

Links

• 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, A000497.

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+3)))/j!); for i from 1 to 20 do S2a(i); od; # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 12 2000

Mathematica

a[n_] := n (n+1) (10n^2+15n+2) (2n+3)!! / 810; Array[a, 20] (* Jean-François Alcover, Feb 09 2016, after Mark van Hoeij *)

Formula

It appears a(n) = 2^(n+1)*GAMMA(n+5/2)*(n^2+n)*(10*n^2+15*n+2)/(405*Pi^(1/2)). - Mark van Hoeij, Oct 26 2011.

G.f.: x*(7*(5-30*x) * hypergeom([4, 9/2],[],2*x) - 26*hypergeom([3, 7/2],[],2*x))/9. - Mark van Hoeij, Apr 07 2013

(n-1)*(10*n^2-5*n-3)*a(n) - (2*n+3)*(n+1)*(10*n^2+15*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