A388131 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n,k) * binomial(4*n+1,k).

1, 7, 76, 918, 11644, 151911, 2018524, 27168952, 369198844, 5054138796, 69594790104, 962888075782, 13374947582676, 186404414887762, 2605298813378472, 36502985628301388, 512546111313760476, 7210406358415710708, 101605251331626574288, 1433918194627756904632

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..700

Formula

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

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

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

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

Mathematica

Table[Sum[ 2^(n-k)* Binomial[ n, k]*Binomial[4*n+1, k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Sep 24 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(n, k)*binomial(4*n+1, k));
(Magma) [&+[2^(n-k)*Binomial(n, k)*Binomial(4*n+1, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 24 2025

Crossrefs

Cf. A388129, A388130.

Cf. A079589, A388135.

Cf. A385605.

Sequence in context: A139472 A180356 A114470 * A098497 A385474 A366015

Adjacent sequences: A388128 A388129 A388130 * A388132 A388133 A388134

Keyword

nonn

Author

Seiichi Manyama, Sep 14 2025

Status

approved