A379547 Expansion of (1/x) * Series_Reversion( x / ( (1+x)^2 * (1+2*x)^4 ) ).

1, 10, 141, 2318, 41586, 789404, 15588677, 316957910, 6591000606, 139521610540, 2996554128002, 65135251885164, 1430214488595340, 31676376789702720, 706819317765805461, 15874751837921964646, 358585244386746378166, 8141109472248910295708

Offset: 0

Author

Seiichi Manyama, Dec 25 2024

Keywords

• nonn
• changed

Links

• Index entries for reversions of series

Crossrefs

Cf. A003168, A371406, A379546.

Programs

Mathematica

CoefficientList[InverseSeries[Series[x / ( (1+x)^2 * (1+2*x)^4 ),{x,0,18}],x]/x,x] (* Stefano Spezia, Aug 25 2025 *)

PARI

my(N=20, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2*(1+2*x)^4))/x)

PARI

a(n) = sum(k=0, n\2, 2^(n-2*k)*binomial(n+1, k)*binomial(5*(n+1)-k, n-2*k))/(n+1);

Formula

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

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(n+1,k) * binomial(5*(n+1)-k,n-2*k).

a(n) = (1/(n+1)) * [x^n] ( (1+x)^2 * (1+2*x)^4 )^(n+1).