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A336539
G.f. A(x) satisfies A(x) = 1 + x * A(x)^3 * (1 + 2 * A(x)).
3
1, 3, 33, 498, 8691, 164937, 3305868, 68855862, 1475636055, 32327521077, 720713175441, 16298128820568, 372946723698516, 8619565476744156, 200920644131737992, 4718057697038124750, 111505342455507462207, 2650261296098965752669, 63308992564445668959795
OFFSET
0,2
LINKS
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
Column k=3 of A336574.
Sequence in context: A071405 A234526 A243251 * A221147 A355795 A291818
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jul 25 2020
STATUS
approved