A370617 -id:A370617 - 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.

A370619 Coefficient of x^n in the expansion of ( (1-x) / (1-x-x^2) )^(2*n).

Original entry on oeis.org

1, 0, 4, 6, 44, 120, 610, 2114, 9468, 36384, 155644, 626450, 2638994, 10856924, 45565118, 189579786, 796023260, 3333362040, 14022032560, 58960463548, 248542728364, 1048148750060, 4427187324102, 18712146312998, 79177190666034, 335259593600120, 1420797366753600

Offset: 0

Seiichi Manyama, May 01 2024

nonn

Cf. A370617, A370618.

Cf. A368957.

a(n, s=2, t=2, u=2) = 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(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)^2 ). See A368957.

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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