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.

A370186 Coefficient of x^n in the expansion of ( (1+x) * (1+x+x^3)^2 )^n.

Original entry on oeis.org

1, 3, 15, 90, 583, 3913, 26790, 185839, 1301575, 9183681, 65181645, 464858661, 3328503814, 23913207750, 172295708971, 1244484142765, 9008351053031, 65332552755149, 474622993450725, 3453219378684621, 25158758123093013, 183521479226172667, 1340195580366321837
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 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(3*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 A369484.

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.