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%I #19 Mar 18 2017 08:42:44
%S 1,1,7,61,655,8365,123795,2082465,39234195,818242425,18711467775,
%T 465512372325,12516455726775,361666448468325,11176241678476875,
%U 367788214424255625,12840711103211866875,474053962648722080625,18451259976779359104375,755138026289116122778125
%N Values of g-hat_2(n), a sum involving Stirling numbers of the first kind.
%H Alois P. Heinz, <a href="/A261687/b261687.txt">Table of n, a(n) for n = 0..400</a>
%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.
%p a:= n-> (m-> add((-1)^k*binomial(2*n+m, n-k)
%p *combinat[stirling1](n+k, k), k=0..n))(-2):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Sep 06 2015
%p # second Maple program:
%p a:= proc(n) option remember; `if`(n<2, 1, ((2*n-3)*
%p (4*n^3+9*n^2-n-3)*a(n-1))/(4*n^3-3*n^2-7*n+3))
%p end:
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Sep 06 2015
%t a[n_] := Sum[(-1)^k Binomial[2n-2, n-k] StirlingS1[n+k, k], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Mar 18 2017 *)
%o (PARI) a(n) = sum(k=0,n,(-1)^k*binomial(2*n-2,n-k)*stirling(n+k,k,1)); \\ _Michel Marcus_, Mar 18 2017
%Y Cf. A261898.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Sep 06 2015
%E More terms from _Alois P. Heinz_, Sep 06 2015
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