A003235 a(n) = Sum_{k=0..n} (-1)^(n-k) C(n,k)*C(k^2,n).
1, 1, 6, 72, 1322, 32550, 1003632, 37162384, 1605962556, 79330914540, 4409098539560, 272297452742304, 18499002436677336, 1371050716542451672, 110085169034456183232, 9519063815009322326400, 881914870734754844035088, 87154631117420724492647184
Offset: 0
Keywords
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. A346184.
Programs
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Mathematica
Table[Sum[(-1)^(n-k) * Binomial[n,k] * Binomial[k^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) / (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
Extensions
More terms from Sean A. Irvine, Mar 19 2015