A370618 -id:A370618 - cp's OEIS frontend

A370617 Coefficient of x^n in the expansion of 1 / (1-x-x^2)^(2*n).

1, 2, 14, 98, 726, 5522, 42770, 335512, 2656998, 21195944, 170076214, 1371181110, 11098310730, 90128497032, 734008622872, 5992486341248, 49028047353670, 401885885751630, 3299812135410080, 27134786911366212, 223433635272820126, 1842041118321640390

Offset: 0

Seiichi Manyama, Apr 30 2024

nonn

Cf. A370618, A370619.

Cf. A368961.

a(n, s=2, t=2, u=0) = 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(3*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 ). See A368961.

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

A370617 Coefficient of x^n in the expansion of 1 / (1-x-x^2)^(2*n).

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