A388046 a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * binomial(4*n+1,k).

1, 11, 181, 3303, 63241, 1244979, 24957661, 506750351, 10387323409, 214483106715, 4454628476805, 92958547262711, 1947496920813081, 40936074806404163, 862915438951374765, 18234593016462779935, 386148722535444633633, 8192809998140199434283

Offset

0,2

Links

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

Formula

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

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

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*n+k+1,n).

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

Mathematica

Table[Sum[2^k*Binomial[n, k]*Binomial[4*n+1, k], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Sep 19 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, 2^k*binomial(n, k)*binomial(4*n+1, k));
(Magma) [&+[2^k*Binomial(n, k)*Binomial(4*n+1, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Sep 19 2025

Crossrefs

Column k=4 of A388052.

Cf. A002002, A108448, A388045.

Sequence in context: A020456 A036935 A205088 * A241193 A143413 A009118

Adjacent sequences: A388043 A388044 A388045 * A388047 A388048 A388049

Keyword

nonn

Author

Seiichi Manyama, Sep 14 2025

Status

approved