A369486 -id:A369486 - cp's OEIS frontend

A370618 Coefficient of x^n in the expansion of ( (1-x) / (1-x-x^2)^2 )^n.

1, 1, 7, 34, 191, 1071, 6154, 35729, 209455, 1236508, 7341577, 43792112, 262230242, 1575391156, 9490934411, 57316715079, 346875036879, 2103174805035, 12773139313516, 77689736488088, 473160660856361, 2885208137132852, 17612514244078288, 107621658416373752

Offset: 0

Seiichi Manyama, Apr 30 2024

nonn

Cf. A370617, A370619.

Cf. A369486.

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

a(n) = Sum_{k=0..floor(n/2)} binomial(2*n+k-1,k) * binomial(2*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-x^2)^2 / (1-x) ). See A369486.

A369489 Expansion of (1/x) * Series_Reversion( x / (1-x) * (1-x-x^3)^2 ).

Original entry on oeis.org

1, 1, 2, 7, 26, 98, 387, 1589, 6688, 28676, 124880, 550926, 2456831, 11056693, 50152457, 229050621, 1052393802, 4861062466, 22559964766, 105144660498, 491922058878, 2309456782464, 10876596029574, 51372213424194, 243283513468707, 1154929327702775, 5495105429597720

Offset: 0

Seiichi Manyama, Jan 24 2024

nonn

Index entries for reversions of series

Cf. A368962, A368966, A368968, A369011.

Cf. A054514, A369486.

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

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

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

cp's OEIS Frontend

A370618 Coefficient of x^n in the expansion of ( (1-x) / (1-x-x^2)^2 )^n.

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A369489 Expansion of (1/x) * Series_Reversion( x / (1-x) * (1-x-x^3)^2 ).

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