A386773 Expansion of (1/x) * Series_Reversion( x * (1-3*x)^2 / (1+2*x)^3 ).

1, 12, 219, 4778, 115011, 2945982, 78764484, 2172877458, 61393035171, 1767592420152, 51672186100899, 1529632003964688, 45760966837725556, 1381338453812353272, 42020564167060633464, 1286902432396816483218, 39645674268979326240291, 1227773019572168363776092

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..654

Index entries for reversions of series

Formula

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

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

D-finite with recurrence: (11880*n^2 + 35640*n + 26400)*a(n) + (-14797*n^2 - 48159*n - 40964)*a(n + 1) + (1643*n^2 + 9312*n + 13120)*a(n + 2) + (-36*n^2 - 270*n - 504)*a(3 + n) = 0. - Robert Israel, Mar 12 2026

Maple

f:= gfun:-rectoproc({(11880*n^2 + 35640*n + 26400)*a(n) + (-14797*n^2 - 48159*n - 40964)*a(n + 1) + (1643*n^2 + 9312*n + 13120)*a(n + 2) + (-36*n^2 - 270*n - 504)*a(3 + n), a(0) = 1, a(1) = 12, a(2) = 219}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Mar 12 2026

Prog

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

Crossrefs

Cf. A386722.

Sequence in context: A087585 A297503 A214512 * A372171 A220068 A377773

Adjacent sequences: A386770 A386771 A386772 * A386774 A386775 A386776

Keyword

nonn

Author

Seiichi Manyama, Aug 02 2025

Status

approved