A369230 -id:A369230 - cp's OEIS frontend

A369229 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / (1-x+x^2)^2 ).

1, 1, 4, 15, 65, 298, 1429, 7073, 35869, 185403, 973198, 5173644, 27797914, 150715321, 823541564, 4530609391, 25073291597, 139492998775, 779706274423, 4376600956063, 24659875131049, 139424357994344, 790763858547445, 4497788153203946, 25650342635871106

Offset: 0

Seiichi Manyama, Jan 17 2024

nonn

Index entries for reversions of series

Cf. A366049, A369230.

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

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

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

A372412 Coefficient of x^n in the expansion of ( (1-x+x^2) / (1-x) )^(3*n).

Original entry on oeis.org

1, 0, 6, 9, 78, 225, 1293, 4851, 24174, 101583, 480531, 2123913, 9869973, 44669742, 206614827, 946394109, 4378019310, 20189406771, 93556141449, 433284753414, 2011960692003, 9345929458455, 43484293732413, 202453490369727, 943647920498997, 4401470801019600

Offset: 0

Seiichi Manyama, Apr 29 2024

nonn

Cf. A372410, A372411.

Cf. A369230.

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

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n,k) * binomial(n-k-1,n-2*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)^3 / (1-x+x^2)^3 ). See A369230.

cp's OEIS Frontend

A369229 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / (1-x+x^2)^2 ).

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A372412 Coefficient of x^n in the expansion of ( (1-x+x^2) / (1-x) )^(3*n).

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