A365700 G.f. satisfies A(x) = 1 + x^5*A(x)^3 / (1 - x*A(x)).

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 4, 8, 13, 19, 26, 46, 88, 163, 284, 466, 781, 1369, 2468, 4449, 7856, 13724, 24084, 42788, 76759, 137785, 246418, 439757, 786132, 1411148, 2541368, 4581906, 8259500, 14889781, 26871106, 48573823, 87934175, 159333544, 288857216

Offset: 0

Author

Seiichi Manyama, Sep 16 2023

Keywords

• nonn

Crossrefs

Cf. A212364, A365698, A365699, A365701, A365702.

Programs

Mathematica

terms = 43; A[] = 0; Do[A[x] = 1 + x^5*A[x]^3 / (1 - x*A[x])+ O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, May 29 2025 *)

PARI

a(n) = sum(k=0, n\5, binomial(n-4*k-1, n-5*k)*binomial(n-2*k+1, k)/(n-2*k+1));

Formula

a(n) = Sum_{k=0..floor(n/5)} binomial(n-4*k-1,n-5*k) * binomial(n-2*k+1,k) / (n-2*k+1).

a(n) ~ s*sqrt((1 - r*s)*(5 - 4*r*s)/(Pi*(3 - r*s*(3 - r*s)))) / (2*n^(3/2)*r^n), where r = 0.53247307479161512230023149440436598140650951738583 and s = 1.2504652351088857309836364363044636883260447207988... are roots of the system of equations r^5*s^3 = (s-1)*(1 - r*s), (s-1)*(3 - 2*r*s) = s*(1 - r*s). - Vaclav Kotesovec, May 29 2025