A336540 G.f. A(x) satisfies A(x) = 1 + x * A(x)^4 * (2 + A(x)).

1, 3, 39, 705, 14799, 338430, 8181597, 205655304, 5320853535, 140761481673, 3790170529806, 103531954949526, 2861975199328581, 79913364673955880, 2250605304332901048, 63855671824327590480, 1823518942311678061503, 52371578117120237562459, 1511737669565948867948805

Offset: 0

Author

Seiichi Manyama, Jul 25 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..671

Crossrefs

Column k=4 of A336575.

Programs

Mathematica

a[n_] := Sum[2^(n-k) * Binomial[n, k] * Binomial[4*n + k + 1, n]/(4*n + k + 1), {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Jul 28 2020 *)

PARI

a(n) = my(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^4*(2+A)); polcoeff(A, n);

PARI

a(n) = if(n==0, 1, sum(k=1, n, 3^k*binomial(n, k)*binomial(4*n, k-1)/n));

PARI

a(n) = sum(k=0, n, 2^(n-k)*binomial(n, k)*binomial(4*n+k+1, n)/(4*n+k+1)); \\ Seiichi Manyama, Jul 28 2020

PARI

a(n) = sum(k=0, n, 2^k*binomial(4*n+1, k)*binomial(5*n-k, n-k))/(4*n+1); \\ Seiichi Manyama, Jul 28 2020

Formula

a(n) = (1/n) * Sum_{k=1..n} 3^k * binomial(n,k) * binomial(4*n,k-1) for n > 0.

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

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

a(n) ~ sqrt(59601 + 205733*sqrt(3/43)) * (7781 + 731*sqrt(129))^(n - 1/2) / (sqrt(Pi) * n^(3/2) * 2^(9*n + 7/2)). - Vaclav Kotesovec, Jul 31 2021

a(n) = (1/n) * Sum_{k=0..n-1} (-2)^k * 3^(n-k) * binomial(n,k) * binomial(5*n-k,n-1-k) for n > 0. - Seiichi Manyama, Aug 10 2023

a(n) = 2^n*binomial(1+4*n, n)*hypergeom([-n, 1+4*n], [2+3*n], -1/2)/(1 + 4*n). - Stefano Spezia, Aug 09 2025