A349427 a(n) = ((n+1)^2 - 2) * binomial(2*n-2,n-1) / 2.

0, 1, 7, 42, 230, 1190, 5922, 28644, 135564, 630630, 2892890, 13117676, 58903572, 262303132, 1159666900, 5094808200, 22259364120, 96773942790, 418882316490, 1805951924700, 7758285404100, 33221013445620, 141830949914940, 603876402587640, 2564713671647400

Offset

0,3

Links

Table of n, a(n) for n=0..24.

Formula

G.f.: x * (1 - x) * (1 - 2*x) / (1 - 4*x)^(5/2).

E.g.f.: x * exp(2*x) * (2 * (1 + x) * BesselI(0,2*x) + (1 + 2*x) * BesselI(1,2*x)) / 2.

a(n) = n * ((n+1)^2 - 2) * Catalan(n-1) / 2.

a(n) = Sum_{k=0..n} binomial(n,k)^2 * A000217(k).

a(n) ~ 2^(2*n-3) * n^(3/2) / sqrt(Pi).

D-finite with recurrence (n-1)*(n^2-2)*a(n) -2*(2*n-3)*(n^2+2*n-1)*a(n-1)=0. - R. J. Mathar, Mar 06 2022

Mathematica

Table[((n + 1)^2 - 2) Binomial[2 n - 2, n - 1]/2, {n, 0, 24}]
nmax = 24; CoefficientList[Series[x (1 - x) (1 - 2 x)/(1 - 4 x)^(5/2), {x, 0, nmax}], x]

Prog

(PARI) a(n) = ((n+1)^2 - 2) * binomial(2*n-2, n-1) / 2 \\ Andrew Howroyd, Nov 20 2021

Crossrefs

Cf. A000108, A000217, A000984, A001700, A001793, A002457, A002544, A008865, A037966, A088218, A127736.

Sequence in context: A374506 A261482 A215226 * A200864 A279613 A162744

Adjacent sequences: A349424 A349425 A349426 * A349428 A349429 A349430

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Nov 17 2021

Status

approved