A336538 G.f. A(x) satisfies A(x) = 1 + x * A(x)^3 * (2 + A(x)).

1, 3, 30, 408, 6402, 109137, 1964010, 36718680, 706221210, 13883562732, 277730910840, 5635185129696, 115693119210270, 2398955889524934, 50167967688522012, 1056869531313301200, 22407983968252808586, 477791976566108489700, 10238908702033904618856, 220401923906465000263200, 4763512100782704414532296

Offset: 0

Author

Seiichi Manyama, Jul 25 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..734

Crossrefs

Column k=3 of A336575.

Programs

Mathematica

a[n_] := Sum[2^(n-k) * Binomial[n, k] * Binomial[3*n + k + 1, n]/(3*n + k + 1), {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Jul 28 2020 *)

PARI

a(n) = my(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^3*(2+A)); polcoeff(A, n);

PARI

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

PARI

a(n) = sum(k=0, n, 2^(n-k)*binomial(n, k)*binomial(3*n+k+1, n)/(3*n+k+1)); \\ Seiichi Manyama, Jul 28 2020

PARI

a(n) = sum(k=0, n, 2^k*binomial(3*n+1, k)*binomial(4*n-k, n-k))/(3*n+1); \\ Seiichi Manyama, Jul 28 2020

Formula

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

a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n,k) * binomial(3*n+k+1,n)/(3*n+k+1).

a(n) = (1/(3*n+1)) * Sum_{k=0..n} 2^k * binomial(3*n+1,k) * binomial(4*n-k,n-k).

a(n) ~ (12 + 8*sqrt(2))^n / (2^(3/4) * sqrt(3*Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 31 2021

a(n) = (1/n) * Sum_{k=0..n-1} (-2)^k * 3^(n-k) * binomial(n,k) * binomial(4*n-k,n-1-k) for n > 0. - Seiichi Manyama, Aug 10 2023

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