A000504 S2(j,2j+3) where S2(n,k) is a 2-associated Stirling number of the second kind. (Formerly M5315 N2309)

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

Offset

1,2

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

Table of n, a(n) for n=1..17.

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.

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

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 *)

Crossrefs

Cf. A008299, A000497.

Sequence in context: A140406 A075512 A223958 * A130646 A038649 A004375

Adjacent sequences: A000501 A000502 A000503 * A000505 A000506 A000507

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved