A371434 Expansion of (1/x) * Series_Reversion( x * ((1-x)^3 + x^2) ).

1, 3, 14, 76, 444, 2702, 16840, 106389, 676566, 4307754, 27333384, 172040544, 1068547788, 6505380072, 38446578888, 217095119499, 1136270066490, 5125578859296, 15040187738184, -47786768551830, -1453703259770520, -18311261482519860, -186896777677695720

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..1125

Index entries for reversions of series

Formula

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

D-finite with recurrence: (-2048*n^3 - 7168*n^2 - 8064*n - 2880)*a(n) + (16640*n^3 + 86400*n^2 + 149328*n + 86040)*a(n + 1) + (-13440*n^3 - 111360*n^2 - 308668*n - 286644)*a(n + 2) + (2648*n^3 + 27184*n^2 + 92312*n + 103776)*a(n + 3) + (-155*n^3 - 1860*n^2 - 7285*n - 9300)*a(n + 4) = 0. - Robert Israel, Mar 11 2026

Maple

f:=  gfun:-rectoproc({(-2048*n^3 - 7168*n^2 - 8064*n - 2880)*a(n) + (16640*n^3 + 86400*n^2 + 149328*n + 86040)*a(n + 1) + (-13440*n^3 - 111360*n^2 - 308668*n - 286644)*a(n + 2) + (2648*n^3 + 27184*n^2 + 92312*n + 103776)*a(n + 3) + (-155*n^3 - 1860*n^2 - 7285*n - 9300)*a(n + 4), a(0) = 1, a(1) = 3, a(2) = 14, a(3) = 76}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Mar 11 2026

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, (-1)^k*binomial(n+k, k)*binomial(4*n+k+2, n-2*k))/(n+1);

Crossrefs

Cf. A371433, A371435.

Cf. A369694.

Sequence in context: A100937 A371785 A223026 * A364477 A364758 A353253

Adjacent sequences: A371431 A371432 A371433 * A371435 A371436 A371437

Keyword

sign

Author

Seiichi Manyama, Mar 23 2024

Status

approved