A000497 S2(j,2j+2) where S2(n,k) is a 2-associated Stirling number of the second kind. (Formerly M5186 N2254)

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

Offset

1,2

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.

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

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 *)

Crossrefs

Cf. A008299, A000504.

Sequence in context: A059946 A357147 A118445 * A353116 A028341 A282689

Adjacent sequences: A000494 A000495 A000496 * A000498 A000499 A000500

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

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

Status

approved