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

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Last modified March 28 05:31 EDT 2023. Contains 361577 sequences. (Running on oeis4.)