A369694 Expansion of (1/x) * Series_Reversion( x * ((1-x)^3-x^2) ).

1, 3, 16, 106, 786, 6244, 51964, 447201, 3947306, 35538668, 325098696, 3013060258, 28232408848, 267003169668, 2545341982728, 24433290332007, 235967943943224, 2291147902820524, 22352525061549604, 219006814853751540, 2154083325737401740

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..976

Index entries for reversions of series

Formula

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

a(n) ~ sqrt((60 + (220324 - 42734*sqrt(2))^(1/3) + (220324 + 42734*sqrt(2))^(1/3)) / (138*Pi)) * (((4/23)*(22 + 3*(293 - 92*sqrt(2))^(1/3) + 3*(293 + 92*sqrt(2))^(1/3)))^n / n^(3/2)). - Vaclav Kotesovec, Jan 29 2024

D-finite with recurrence: 32*(4*n + 5)*(2*n + 3)*(4*n + 3)*a(n) + 8*(104*n^3 + 1080*n^2 + 3072*n + 2625)*a(n + 1) - 4*(n + 3)*(156*n^2 + 1596*n + 3031)*a(n + 2) + 4*(n + 4)*(n + 3)*(439*n + 1594)*a(n + 3) - 161*(n + 5)*(n + 4)*(n + 3)*a(n + 4) = 0. - Robert Israel, May 01 2026

Maple

f:= gfun:-rectoproc({32*(4*n + 5)*(2*n + 3)*(4*n + 3)*a(n) + 8*(104*n^3 + 1080*n^2 + 3072*n + 2625)*a(n + 1) - 4*(n + 3)*(156*n^2 + 1596*n + 3031)*a(n + 2) + 4*(n + 4)*(n + 3)*(439*n + 1594)*a(n + 3) - 161*(n + 5)*(n + 4)*(n + 3)*a(n + 4), a(0) = 1, a(1) = 3, a(2) = 16, a(3) = 106}, a(n), remember):
map(f, [$0..30]); # Robert Israel, May 01 2026

Mathematica

CoefficientList[InverseSeries[Series[x*((1-x)^3 - x^2), {x, 0, 30}], x]/x, x](* Vaclav Kotesovec, Jan 29 2024 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^3-x^2))/x)
(PARI) a(n) = sum(k=0, n\2, binomial(n+k, k)*binomial(4*n+k+2, n-2*k))/(n+1);

Crossrefs

Cf. A001002, A151374.

Cf. A369114, A369161, A369215.

Sequence in context: A370092 A063548 A157452 * A381879 A074551 A135074

Adjacent sequences: A369691 A369692 A369693 * A369695 A369696 A369697

Keyword

nonn

Author

Seiichi Manyama, Jan 29 2024

Status

approved