OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..350
FORMULA
a(n) = [x^n] (1+x)^(3*n+1)/(1-2*x)^(2*n+1).
a(n) = [x^n] 1/((1-x) * (1-3*x)^(2*n+1)).
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(3*n+1,k).
a(n) = Sum_{k=0..n} 3^k * binomial(2*n+k,k).
MATHEMATICA
Table[Sum[2^(n-k)*Binomial[3*n+1, k]*Binomial[3*n-k, n-k], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Aug 05 2025 *)
PROG
(PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(3*n+1, k)*binomial(3*n-k, n-k));
(Magma) [&+[2^(n-k) * Binomial(3*n+1, k) * Binomial(3*n-k, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 05 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Aug 04 2025
STATUS
approved