A003235 a(n) = Sum_{k=0..n} (-1)^(n-k) C(n,k)*C(k^2,n). (Formerly M4280)

1, 1, 6, 72, 1322, 32550, 1003632, 37162384, 1605962556, 79330914540, 4409098539560, 272297452742304, 18499002436677336, 1371050716542451672, 110085169034456183232, 9519063815009322326400, 881914870734754844035088, 87154631117420724492647184

Offset

0,3

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) / (2*Pi*sqrt(2*(1-w)/w)) = 0.150381859108542022051646532351211728293419626579836320368956458003898775818..., where w = -LambertW(-2*exp(-2)) = -A226775. - Vaclav Kotesovec, Dec 13 2020, updated Jul 09 2021

A003236(n) / a(n) ~ -2 / LambertW(-2*exp(-2)) = 4.92155363456750509... - Vaclav Kotesovec, Jul 09 2021

Mathematica

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

Crossrefs

Cf. A346184.

Sequence in context: A202382 A266869 A001763 * A113133 A302355 A089252

Adjacent sequences: A003232 A003233 A003234 * A003236 A003237 A003238

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Mar 19 2015

Status

approved