A369503 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2+x^2)^2 ).

1, 4, 24, 168, 1284, 10384, 87360, 756704, 6703168, 60444928, 552990592, 5120101760, 47887472000, 451759449600, 4293634467840, 41073654689280, 395170166443008, 3821262491103232, 37118973530660864, 362035991963869184, 3544080121528001536

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..978

Index entries for reversions of series

Formula

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

D-finite with recurrence: (2304*n^3 + 11520*n^2 + 17856*n + 8640)*a(n) + (4768*n^3 + 30672*n^2 + 65024*n + 45360)*a(n + 1) + (-758*n^3 - 6744*n^2 - 20038*n - 19932)*a(n + 2) + (27*n^3 + 324*n^2 + 1293*n + 1716)*a(n + 3) = 0. - Robert Israel, Apr 20 2026

Maple

f:= gfun:-rectoproc({(2304*n^3 + 11520*n^2 + 17856*n + 8640)*a(n) + (4768*n^3 + 30672*n^2 + 65024*n + 45360)*a(n + 1) + (-758*n^3 - 6744*n^2 - 20038*n - 19932)*a(n + 2) + (27*n^3 + 324*n^2 + 1293*n + 1716)*a(n + 3), a(0) = 1, a(1) = 4, a(2) = 24}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Apr 20 2026

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2+x^2)^2)/x)
(PARI) a(n) = sum(k=0, n\2, binomial(2*n+2, k)*binomial(4*n-2*k+4, n-2*k))/(n+1);

Crossrefs

Cf. A369502, A369504.

Sequence in context: A339346 A214377 A331007 * A212277 A246423 A188913

Adjacent sequences: A369500 A369501 A369502 * A369504 A369505 A369506

Keyword

nonn

Author

Seiichi Manyama, Jan 25 2024

Status

approved