cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A370185 Coefficient of x^n in the expansion of (1+x+x^3)^(2*n).

Original entry on oeis.org

1, 2, 6, 26, 126, 612, 2970, 14534, 71838, 357884, 1793296, 9026976, 45612450, 231224060, 1175422590, 5989693176, 30586693182, 156483812892, 801908994852, 4115509738188, 21149522157816, 108817959549416, 560500440662872, 2889915938877078, 14913928051929426
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(2*n,k) * binomial(2*n-k,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+x^3)^2 ). See A369483.

A370187 Coefficient of x^n in the expansion of ( (1+x)^2 * (1+x+x^3)^2 )^n.

Original entry on oeis.org

1, 4, 28, 226, 1940, 17214, 155914, 1432106, 13289076, 124276528, 1169346298, 11057293526, 104986087178, 1000248093420, 9557756114130, 91559051752596, 879027678226452, 8455595252761536, 81476137225450096, 786286875175380088, 7598503022428758570
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(2*n,k) * binomial(4*n-k,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)^2 * (1+x+x^3)^2) ). See A369485.

A372371 Coefficient of x^n in the expansion of ( (1+x+x^3)^2 / (1+x) )^n.

Original entry on oeis.org

1, 1, 1, 7, 25, 61, 187, 666, 2137, 6676, 22001, 73217, 239923, 789517, 2624182, 8729527, 29026553, 96790606, 323546416, 1082566763, 3626148425, 12163438539, 40847087821, 137294721676, 461890741843, 1555264438186, 5240857508017, 17672768973979, 59634361740734
Offset: 0

Views

Author

Seiichi Manyama, Apr 28 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(2*n,k) * binomial(n-k,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)^2 ). See A372375.
Showing 1-3 of 3 results.