A003236 a(n) = Sum_{k=0..n} (-1)^(n-k) C(n,k)*C((k+1)^2, n).

1, 3, 24, 320, 6122, 153762, 4794664, 178788528, 7762727196, 384733667780, 21434922419504, 1326212860090560, 90227121642144424, 6694736236093168200, 538028902298395832832, 46558260925421295229568, 4316186393637505403773328

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

• H. W. Gould, personal communication.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300
• Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974.

Crossrefs

Cf. A346183.

Programs

Mathematica

Table[Sum[(-1)^(n-k) * Binomial[n,k] * Binomial[(k+1)^2, n], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Dec 13 2020 *)

Formula

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

a(n) / A003235(n) ~ -2 / LambertW(-2*exp(-2)) = 4.92155363456750509... - Vaclav Kotesovec, Jul 09 2021

Extensions

More terms from Sean A. Irvine, Mar 19 2015