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%I M5315 N2309 #33 Jul 03 2018 21:16:25
%S 1,56,1918,56980,1636635,47507460,1422280860,44346982680,
%T 1446733012725,49473074851200,1774073543492250,66681131440423500,
%U 2624634287988087375,108060337458000427500,4647703259223579555000,208548093035794902390000,9749651260035434678555625
%N S2(j,2j+3) where S2(n,k) is a 2-associated Stirling number of the second kind.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 296.
%H H. W. Gould, Harris Kwong, Jocelyn Quaintance, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Kwong/kwong9.html">On Certain Sums of Stirling Numbers with Binomial Coefficients</a>, J. Integer Sequences, 18 (2015), #15.9.6.
%H M. Ward, <a href="http://www.jstor.org/stable/2370916">The representations of Stirling's numbers and Stirling's polynomials as sums of factorials</a>, Amer. J. Math., 56 (1934), p. 87-95.
%F 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.
%F 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
%F (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
%p 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
%t 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_ *)
%Y Cf. A008299, A000497.
%K nonn
%O 1,2
%A _N. J. A. Sloane_
%E More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 12 2000
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