A387933 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n,k) * binomial(4*n,k).

1, 7, 85, 1165, 16829, 250422, 3798199, 58380840, 906219405, 14173518325, 223009006090, 3525971612830, 55973720349695, 891583186291480, 14242824447215400, 228094787567204880, 3660841476461283885, 58867980292782225855, 948234279492305885935, 15297177348002010962803

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Formula

a(n) = [x^n] (1+2*x)^n/(1-x)^(4*n+1).

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

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

a(n) = [x^n] ((1+x)^4 * (1+3*x))^n.

a(n) ~ (38343 + 5035*sqrt(57))^n / (sqrt((41*sqrt(57) - 285)*Pi*n) * 2^(9*n-2) * 3^(2*n)). - Vaclav Kotesovec, Sep 21 2025

Mathematica

Table[Sum[ 3^(n-k)*Binomial[ n, k]*Binomial[4*n, k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Sep 20 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, 3^(n-k)*binomial(n, k)*binomial(4*n, k));
(Magma) [&+[3^(n-k)*Binomial(n, k)*Binomial(4*n, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Sep 20 2025

Crossrefs

Cf. A387929, A387931.

Sequence in context: A193578 A309187 A026001 * A388726 A371363 A386397

Adjacent sequences: A387930 A387931 A387932 * A387934 A387935 A387936

Keyword

nonn

Author

Seiichi Manyama, Sep 13 2025

Status

approved