OFFSET
0,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..710
FORMULA
a(n) = Sum_{k=0..n} 2^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^(n-k) * binomial(3*n+1,k) * binomial(4*n-k,n-k).
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
From Seiichi Manyama, Aug 10 2023: (Start)
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.
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)
MATHEMATICA
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 *)
PROG
(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);
(PARI) a(n) = sum(k=0, n, 2^k*binomial(n, k)*binomial(3*n+k+1, n)/(3*n+k+1));
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jul 25 2020
STATUS
approved