A370172 -id:A370172 - cp's OEIS frontend

A370170 Coefficient of x^n in the expansion of (1+x+x^2)^(3*n).

1, 3, 21, 156, 1221, 9828, 80580, 669294, 5612805, 47419905, 402993396, 3441242544, 29502452868, 253778827695, 2189249293266, 18932541179706, 164081616775173, 1424741956592535, 12392093363519415, 107946143556797700, 941580123046540596

Offset: 0

Seiichi Manyama, Feb 11 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A370171, A370172.

Cf. A365128.

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

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n,k) * binomial(3*n-k,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)^3 ). See A365128.

A370171 Coefficient of x^n in the expansion of ( (1+x) * (1+x+x^2)^3 )^n.

Original entry on oeis.org

1, 4, 34, 319, 3146, 31929, 330145, 3458620, 36585194, 389893576, 4179819559, 45025583343, 486961123577, 5284324727023, 57508473997848, 627410367071169, 6859805605391466, 75144918246760324, 824558759018846116, 9061483047671168437, 99716283188165243471

Offset: 0

Seiichi Manyama, Feb 11 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..951

Cf. A370170, A370172.

Cf. A369479.

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

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n,k) * binomial(4*n-k,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) * (1+x+x^2)^3) ). See A369479.

A372369 Coefficient of x^n in the expansion of ( (1+x+x^2)^3 / (1+x) )^n.

Original entry on oeis.org

1, 2, 12, 65, 388, 2352, 14565, 91289, 577764, 3683459, 23621462, 152203482, 984598741, 6390596591, 41596873869, 271424778015, 1774892605284, 11628321367815, 76311803660025, 501554760288813, 3300889231760238, 21750690436059188, 143481522241226962

Offset: 0

Seiichi Manyama, Apr 28 2024

nonn

Cf. A351858, A370170, A370171, A370172.

Cf. A347953.

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

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

cp's OEIS Frontend

A370170 Coefficient of x^n in the expansion of (1+x+x^2)^(3*n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A370171 Coefficient of x^n in the expansion of ( (1+x) * (1+x+x^2)^3 )^n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A372369 Coefficient of x^n in the expansion of ( (1+x+x^2)^3 / (1+x) )^n.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula