A331552 Expansion of (1 + 2*x)/(1 + 4*x^2)^(3/2).

1, 2, -6, -12, 30, 60, -140, -280, 630, 1260, -2772, -5544, 12012, 24024, -51480, -102960, 218790, 437580, -923780, -1847560, 3879876, 7759752, -16224936, -32449872, 67603900, 135207800, -280816200, -561632400, 1163381400, 2326762800, -4808643120, -9617286240, 19835652870

Offset: 0

Author

Seiichi Manyama, Jan 20 2020

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Column 1 of A331511.

Cf. A100071.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 33); Coefficients(R!( (1 + 2*x)/(1 + 4*x^2)^(3/2))); // Marius A. Burtea, Jan 20 2020

Magma

[&+[(-1)^k*(k+1)*Binomial(n+1, k+1)^2:k in [0..n]]:n in [0..33]]; // Marius A. Burtea, Jan 20 2020

Mathematica

a[n_] := Sum[(-1)^k * (k + 1) * Binomial[n + 1, k + 1]^2, {k, 0, n}]; Array[a, 33, 0] (* Amiram Eldar, Jan 20 2020 *)

PARI

N=66; x='x+O('x^N); Vec((1+2*x)/(1+4*x^2)^(3/2))

PARI

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

PARI

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

Formula

|a(n)| = A100071(n+1).

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

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

n * (2*n-1) * a(n) = 2 * a(n-1) - 4 * n * (2*n+1) * a(n-2) for n>1.

E.g.f.: (1 + 2*x)*BesselJ(0,2*x) - 2*x*BesselJ(1,2*x). - Ilya Gutkovskiy, Mar 04 2021