A385474 Expansion of (1/x) * Series_Reversion( x * (1-2*x)^2 / (1+x)^3 ).

1, 7, 76, 991, 14281, 219172, 3512440, 58096591, 984340003, 16996883887, 298017184048, 5291703108292, 94961611382860, 1719543577996888, 31379622840361696, 576519956457976495, 10655055147825932119, 197959348525977645781, 3695112941037246866044

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..765

Index entries for reversions of series

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} 2^(n-k) * binomial(3*(n+1),k) * binomial(3*n-k+1,n-k).

a(n) = (1/(n+1)) * [x^n] ( (1+x)^3 / (1-2*x)^2 )^(n+1).

D-finite with recurrence: 189*(3*n + 5)*(3*n + 4)*a(n) - 6*(1202*n^2 + 3789*n + 3066)*a(n + 1) + 16*(103*n^2 + 588*n + 833)*a(n + 2) - 32*(n + 4)*(2*n + 7)*a(n + 3) = 0. - Robert Israel, Jun 07 2026

Maple

f:= gfun:-rectoproc({189*(3*n + 5)*(3*n + 4)*a(n) - 6*(1202*n^2 + 3789*n + 3066)*a(n + 1) + 16*(103*n^2 + 588*n + 833)*a(n + 2) - 32*(n + 4)*(2*n + 7)*a(n + 3), a(0) = 1, a(1) = 7, a(2) = 76}, a(n), remember):
map(f, [$0..20]); # Robert Israel, Jun 07 2026

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-2*x)^2/(1+x)^3)/x)
(PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(3*(n+1), k)*binomial(3*n-k+1, n-k))/(n+1);

Crossrefs

Cf. A064063, A385475.

Cf. A384950.

Sequence in context: A114470 A388131 A098497 * A366015 A386535 A380972

Adjacent sequences: A385471 A385472 A385473 * A385475 A385476 A385477

Keyword

nonn,changed

Author

Seiichi Manyama, Aug 01 2025

Status

approved