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A000497 S2(j,2j+2) where S2(n,k) is a 2-associated Stirling number of the second kind.
(Formerly M5186 N2254)
2
1, 25, 490, 9450, 190575, 4099095, 94594500, 2343240900, 62199262125, 1764494857125, 53338158823950, 1712934942468750, 58274046742786875, 2094379201311271875, 79318164037837725000, 3157886388887074845000 (list; graph; refs; listen; history; text; internal format)
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

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-Fran├žois 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: A014927 A059946 A118445 * A028341 A282689 A282874

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

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Last modified August 18 14:29 EDT 2017. Contains 290720 sequences.