A003236 a(n) = Sum_{k=0..n} (-1)^(n-k) C(n,k)*C((k+1)^2, n). (Formerly M3107)

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

Offset

0,2

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.

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

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 *)

Crossrefs

Cf. A346183.

Sequence in context: A047056 A343445 A264561 * A232693 A319939 A375869

Adjacent sequences: A003233 A003234 A003235 * A003237 A003238 A003239

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Mar 19 2015

Status

approved