A349018 G.f. A(x) satisfies A(x) = 1/(1 - x/(1 - x*A(x)))^4.

1, 4, 14, 60, 297, 1584, 8868, 51412, 305964, 1858308, 11472152, 71774548, 454080514, 2899959640, 18670920458, 121056521536, 789733186076, 5180002637472, 34141018474400, 225995779077324, 1501809350268648, 10015202238242356, 67003372168525774

Offset: 0

Author

Seiichi Manyama, Nov 06 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A262441, A349017.

Cf. A364723.

Programs

PARI

a(n, s=1, t=4) = sum(k=0, n, binomial(t*n-(t-1)*(k-1), k)*binomial(n+(s-1)*k-1, n-k)/(n-k+1));

Formula

If g.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^s)^t, then a(n) = Sum_{k=0..n} binomial(t*n-(t-1)*(k-1),k) * binomial(n+(s-1)*k-1,n-k)/(n-k+1).

a(n) ~ sqrt((1 - r*s)*(1 - r - r*s) / (2 - r*(2*s - 3))) / (sqrt(2*Pi) * n^(3/2) * r^(n+1)), where r = 0.13968480593491705709394976139265608086009606657813769... and s = 3.10146641162846907900664383717504887133026560522911567... are real roots of the system of equations (-1 + r*s)^4/(-1 + r + r*s)^4 = s, (4*r^2*(-1 + r*s)^3)/(-1 + r + r*s)^5 = 1. - Vaclav Kotesovec, Nov 15 2021

G.f.: A(x) = B(x)^4 where B(x) is the g.f. of A364723. - Seiichi Manyama, Dec 04 2024