A369502 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2+x)^2 ).

1, 6, 47, 420, 4059, 41316, 436345, 4737018, 52535950, 592667532, 6779699073, 78458218746, 916886214115, 10805128064100, 128260666769895, 1532180536574580, 18405744106135914, 222204347510440092, 2694506677864591810, 32804976554127379680, 400837173223351237295

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..894

Index entries for reversions of series

Formula

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

D-finite with recurrence: (52000*n^3 + 260000*n^2 + 403000*n + 195000)*a(n) + (40920*n^3 + 262980*n^2 + 556260*n + 386400)*a(n + 1) + (-4482*n^3 - 38286*n^2 - 107664*n - 99480)*a(n + 2) + (81*n^3 + 972*n^2 + 3879*n + 5148)*a(n + 3) = 0. - Robert Israel, Apr 19 2026

Maple

f:= gfun:-rectoproc({(52000*n^3 + 260000*n^2 + 403000*n + 195000)*a(n) + (40920*n^3 + 262980*n^2 + 556260*n + 386400)*a(n + 1) + (-4482*n^3 - 38286*n^2 - 107664*n - 99480)*a(n + 2) + (81*n^3 + 972*n^2 + 3879*n + 5148)*a(n + 3), a(0) = 1, a(1) = 6, a(2) = 47}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Apr 19 2026

Prog

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

Crossrefs

Cf. A369503, A369504.

Cf. A003645, A369505.

Sequence in context: A391495 A390333 A071878 * A364748 A383558 A365186

Adjacent sequences: A369499 A369500 A369501 * A369503 A369504 A369505

Keyword

nonn

Author

Seiichi Manyama, Jan 25 2024

Status

approved