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a(n) = Sum_{k=0..n} (-1)^(n-k) C(n,k)*C((k+1)^2, n).
(Formerly M3107)
3

%I M3107 #33 Dec 26 2021 20:41:30

%S 1,3,24,320,6122,153762,4794664,178788528,7762727196,384733667780,

%T 21434922419504,1326212860090560,90227121642144424,

%U 6694736236093168200,538028902298395832832,46558260925421295229568,4316186393637505403773328

%N a(n) = Sum_{k=0..n} (-1)^(n-k) C(n,k)*C((k+1)^2, n).

%D H. W. Gould, personal communication.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A003236/b003236.txt">Table of n, a(n) for n = 0..300</a>

%H Henry W. Gould, <a href="/A003099/a003099.pdf">Letters to N. J. A. Sloane, Oct 1973 and Jan 1974</a>.

%F a(n) ~ c * d^n * (n-1)!, where d = 4 / (w*(2-w)) = 6.17655460948348035823168... and c = exp(1/2 - w^2/8) / (Pi*sqrt(2*w*(1-w))) = 0.740112385268663459927202070799244309431121698475089032623558890186368006364..., where w = -LambertW(-2*exp(-2)) = -A226775. - _Vaclav Kotesovec_, Dec 13 2020, updated Jul 09 2021

%F a(n) / A003235(n) ~ -2 / LambertW(-2*exp(-2)) = 4.92155363456750509... - _Vaclav Kotesovec_, Jul 09 2021

%t Table[Sum[(-1)^(n-k) * Binomial[n,k] * Binomial[(k+1)^2, n], {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Dec 13 2020 *)

%Y Cf. A346183.

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Sean A. Irvine_, Mar 19 2015