A361699 Constant term in the expansion of (1 + x^3 + y^3 + 1/(x*y))^n.

1, 1, 1, 1, 1, 21, 121, 421, 1121, 2521, 6301, 23101, 99001, 386101, 1301301, 3943941, 11779041, 38241841, 136988041, 504616441, 1793870941, 6061831441, 19923689941, 66139128441, 227052188441, 800641330721, 2831644750221, 9870443816221, 33869987735221

Offset

0,6

Links

Winston de Greef, Table of n, a(n) for n = 0..1798

Formula

a(n) = Sum_{k=0..floor(n/5)} binomial(2*k,k) * binomial(5*k,2*k) * binomial(n,5*k).

From Vaclav Kotesovec, Mar 22 2023: (Start)

Recurrence: 3*n^2*(3*n - 10)*(3*n - 5)*a(n) = 3*(45*n^4 - 270*n^3 + 510*n^2 - 375*n + 104)*a(n-1) - 45*(n-1)*(6*n^3 - 36*n^2 + 67*n - 40)*a(n-2) + 15*(n-2)*(n-1)*(18*n^2 - 90*n + 109)*a(n-3) - 135*(n-3)^2*(n-2)*(n-1)*a(n-4) + 3152*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).

a(n) ~ sqrt(c) * (1 + 5/3^(3/5))^n / (Pi * n), where c = 0.8011502211360696582191471740430432783906089377204901279920664641344364478... is the real root of the equation -2483776 + 28284375*c - 141840000*c^2 + 337500000*c^3 - 405000000*c^4 + 194400000*c^5 = 0. (End)

Mathematica

Table[Sum[Binomial[2*k, k] * Binomial[5*k, 2*k] * Binomial[n, 5*k], {k, 0, n/5}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 22 2023 *)

Prog

(PARI) a(n) = sum(k=0, n\5, binomial(2*k, k)*binomial(5*k, 2*k)*binomial(n, 5*k));

Crossrefs

Cf. A344560, A361657, A361700.

Sequence in context: A044734 A325484 A365205 * A200888 A179441 A164785

Adjacent sequences: A361696 A361697 A361698 * A361700 A361701 A361702

Keyword

nonn,easy

Author

Seiichi Manyama, Mar 21 2023

Status

approved