A371669 -id:A371669 - cp's OEIS frontend

A371661 G.f. satisfies A(x) = 1 + x * A(x)^3 * (1 + A(x))^2.

1, 4, 64, 1424, 36800, 1036160, 30843648, 954671360, 30415326208, 990831196160, 32853724512256, 1105132250898432, 37620337933582336, 1293586791397064704, 44863864476704768000, 1567543145774827241472, 55125711913212153954304

Offset: 0

Seiichi Manyama, Apr 01 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..633

Cf. A271469, A363311, A371660.

Cf. A371669.

a(n) = if(n==0, 1, sum(k=0, (n-1)\2, 4^(n-k)*binomial(n, k)*binomial(4*n-k, n-1-2*k))/n);

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

a(n) = 2^n * A371669(n). - Seiichi Manyama, Dec 26 2024

A379546 Expansion of (1/x) * Series_Reversion( x / ( (1+x)^2 * (1+2*x)^3 ) ).

Original entry on oeis.org

1, 8, 89, 1150, 16190, 240966, 3729185, 59404934, 967608590, 16041857672, 269807678442, 4592326407908, 78954271935856, 1369136489157250, 23918810207745777, 420575805001923782, 7437459126200243030, 132190772588551036800, 2360148586461490077870

Offset: 0

Seiichi Manyama, Dec 25 2024

nonn

Index entries for reversions of series

Cf. A003168, A371406, A379547.
Cf. A371398, A379522.
Cf. A371669.

my(N=20, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2*(1+2*x)^3))/x)

a(n) = sum(k=0, n\2, 2^(n-2*k)*binomial(n+1, k)*binomial(4*(n+1)-k, n-2*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..n} 2^k * binomial(3*(n+1),k) * binomial(2*(n+1),n-k).
a(n) = A371669(n+1)/2 = (1/(n+1)) * Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(n+1,k) * binomial(4*(n+1)-k,n-2*k).
a(n) = (1/(n+1)) * [x^n] ( (1+x)^2 * (1+2*x)^3 )^(n+1).

cp's OEIS Frontend

A371661 G.f. satisfies A(x) = 1 + x * A(x)^3 * (1 + A(x))^2.

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A379546 Expansion of (1/x) * Series_Reversion( x / ( (1+x)^2 * (1+2*x)^3 ) ).

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