A371772 a(n) = Sum_{k=0..floor(n/3)} binomial(5*n-3*k-1,n-3*k).

1, 4, 36, 365, 3892, 42714, 477621, 5411109, 61901268, 713435333, 8271470666, 96361329024, 1127086021461, 13227336997645, 155680966681101, 1836862248992565, 21719923705450260, 257316706385394615, 3053599633736172765, 36292098436808314572, 431918050456887676362

Offset

0,2

Links

Harvey P. Dale, Table of n, a(n) for n = 0..922

Formula

a(n) = [x^n] 1/((1-x^3) * (1-x)^(4*n)).

a(n) = binomial(5*n-1, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [(1-5*n)/3, (2-5*n)/3, 1-5*n/3], 1). - Stefano Spezia, Apr 06 2024

From Vaclav Kotesovec, Apr 08 2024: (Start)

Recurrence: 72*n*(2*n - 1)*(4*n - 3)*(4*n - 1)*(899*n^2 - 2355*n + 1534)*a(n) = (25514519*n^6 - 117751221*n^5 + 212960873*n^4 - 191684487*n^3 + 89835824*n^2 - 20567076*n + 1769040)*a(n-1) - 5*(5*n - 7)*(5*n - 6)*(5*n - 4)*(5*n - 3)*(899*n^2 - 557*n + 78)*a(n-2).

a(n) ~ 5^(5*n + 5/2) / (31 * sqrt(Pi*n) * 2^(8*n + 3/2)). (End)

a(n) = Sum_{k=0..n} (-3)^k * binomial(5*n+k+2,n-k). - Seiichi Manyama, Nov 13 2025

Mathematica

Table[Sum[Binomial[5n-3k-1, n-3k], {k, 0, Floor[n/3]}], {n, 0, 20}] (* Harvey P. Dale, Mar 31 2026 *)

Prog

(PARI) a(n) = sum(k=0, n\3, binomial(5*n-3*k-1, n-3*k));

Crossrefs

Cf. A371758, A371770, A371771.

Cf. A371756.

Sequence in context: A372465 A168595 A163455 * A138736 A372461 A379022

Adjacent sequences: A371769 A371770 A371771 * A371773 A371774 A371775

Keyword

nonn

Author

Seiichi Manyama, Apr 05 2024

Status

approved