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

%I #37 Aug 10 2023 10:27:54

%S 1,3,33,498,8691,164937,3305868,68855862,1475636055,32327521077,

%T 720713175441,16298128820568,372946723698516,8619565476744156,

%U 200920644131737992,4718057697038124750,111505342455507462207,2650261296098965752669,63308992564445668959795

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

%H Seiichi Manyama, <a href="/A336539/b336539.txt">Table of n, a(n) for n = 0..710</a>

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

%F a(n) ~ sqrt(168 + 97*sqrt(3)) * (26 + 15*sqrt(3))^(n - 1/2) / (3*sqrt(Pi) * n^(3/2) * 2^(n + 3/2)). - _Vaclav Kotesovec_, Jul 31 2021

%F From _Seiichi Manyama_, Aug 10 2023: (Start)

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

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

%t a[n_] := Sum[2^k * Binomial[n, k] * Binomial[3*n + k + 1, n]/(3*n + k + 1), {k, 0, n}]; Array[a, 19, 0] (* _Amiram Eldar_, Jul 27 2020 *)

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

%o (PARI) a(n) = sum(k=0, n, 2^k*binomial(n, k)*binomial(3*n+k+1, n)/(3*n+k+1));

%o (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(3*n+1, k)*binomial(4*n-k, n-k))/(3*n+1); \\ _Seiichi Manyama_, Jul 26 2020

%Y Column k=3 of A336574.

%Y Cf. A243659, A336538.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jul 25 2020