A372414 -id:A372414 - cp's OEIS frontend

A372413 Coefficient of x^n in the expansion of ( (1-x+x^3) / (1-x) )^n.

1, 0, 0, 3, 4, 5, 21, 49, 92, 237, 595, 1331, 3169, 7787, 18487, 44108, 107036, 258349, 622371, 1508239, 3658679, 8869465, 21543005, 52399612, 127497281, 310487855, 756858661, 1846060464, 4505442967, 11003284052, 26887642756, 65735882819, 160795695676

Offset: 0

Seiichi Manyama, Apr 29 2024

nonn

Cf. A372414, A372415.

Cf. A114997.

a(n, s=3, t=1, u=1) = 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/3)} binomial(n,k) * binomial(n-2*k-1,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) / (1-x+x^3) ).

A372415 Coefficient of x^n in the expansion of ( (1-x+x^3) / (1-x)^3 )^n.

Original entry on oeis.org

1, 2, 10, 59, 366, 2332, 15121, 99276, 657894, 4391438, 29482320, 198865680, 1346655921, 9149295482, 62336961732, 425760311734, 2914151872614, 19983724103726, 137267022656710, 944287970305935, 6504676822047876, 44861522295224400, 309742638630690264

Offset: 0

Seiichi Manyama, Apr 29 2024

nonn

Cf. A372413, A372414.

Cf. A366052.

a(n, s=3, t=1, 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/3)} binomial(n,k) * binomial(3*n-2*k-1,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)^3 / (1-x+x^3) ). See A366052.

cp's OEIS Frontend

A372413 Coefficient of x^n in the expansion of ( (1-x+x^3) / (1-x) )^n.

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