A369443 -id:A369443 - cp's OEIS frontend

A370196 Coefficient of x^n in the expansion of ( (1+x)^2 * (1+x^3) )^n.

1, 2, 6, 23, 102, 477, 2259, 10733, 51174, 245156, 1180381, 5709387, 27723315, 135055845, 659744973, 3230479173, 15850993126, 77918426928, 383646423564, 1891715752242, 9340099603677, 46170434726054, 228479085858447, 1131770152854441, 5611302030239667

Offset: 0

Seiichi Manyama, Feb 11 2024

Cf. A369443.

a(n, s=3, t=1, u=2) = sum(k=0, n\s, binomial(t*n, k)*binomial(u*n, n-s*k));

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

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x / ((1+x)^2 * (1+x^3)) ). See A369443.

A368318 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * (1+x^3)^2) ).

Original entry on oeis.org

1, 2, 5, 16, 62, 264, 1170, 5310, 24599, 116090, 556569, 2703098, 13268900, 65721840, 328050639, 1648535856, 8333536002, 42348587700, 216211838178, 1108514508608, 5704874555112, 29460504457692, 152612723209700, 792833380805160, 4129639139612133

Offset: 0

Seiichi Manyama, Jan 23 2024

nonn

Index entries for reversions of series

Cf. A369442, A369443.

my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2*(1+x^3)^2))/x)

a(n, s=3, t=2, u=2) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial(u*(n+1), n-s*k))/(n+1);

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

cp's OEIS Frontend

A370196 Coefficient of x^n in the expansion of ( (1+x)^2 * (1+x^3) )^n.

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A368318 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * (1+x^3)^2) ).

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