A336538 - OEIS

%I #27 Aug 10 2023 10:27:40

%S 1,3,30,408,6402,109137,1964010,36718680,706221210,13883562732,

%T 277730910840,5635185129696,115693119210270,2398955889524934,

%U 50167967688522012,1056869531313301200,22407983968252808586,477791976566108489700,10238908702033904618856,220401923906465000263200,4763512100782704414532296

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

%H Seiichi Manyama, <a href="/A336538/b336538.txt">Table of n, a(n) for n = 0..734</a>

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

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

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

%F a(n) ~ (12 + 8*sqrt(2))^n / (2^(3/4) * sqrt(3*Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jul 31 2021

%F 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

%t 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 *)

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

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

%o (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

%o (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

%Y Column k=3 of A336575.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jul 25 2020