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.

A385320 a(n) = Sum_{k=0..n} 2^k * binomial(3*n,k) * binomial(3*n-k-1,n-k).

Original entry on oeis.org

1, 8, 118, 1970, 34714, 630548, 11678284, 219240008, 4157096266, 79429466456, 1526869550638, 29495424821354, 572100064904872, 11134578632483600, 217341014671302976, 4253067310380772400, 83409477100625759050, 1638952453699219007072, 32259670449587082804466
Offset: 0

Views

Author

Seiichi Manyama, Jul 31 2025

Keywords

Crossrefs

Cf. A385319, A386719.
Cf. A386722.

Programs

  • Mathematica
    Table[Sum[3^k*(-1)^(n-k)*Binomial[3*n, k], {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jul 31 2025 *)
  • PARI
    a(n) = sum(k=0, n, 2^k*binomial(3*n, k)*binomial(3*n-k-1, n-k));

Formula

a(n) = [x^n] ( (1+2*x)^3/(1-x)^2 )^n.
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+2*x)^3 ). See A386722.
a(n) = Sum_{k=0..n} 3^k * (-1)^(n-k) * binomial(3*n,k).
a(n) ~ 3^(4*n + 3/2) / (7*sqrt(Pi*n)*2^(2*n)). - Vaclav Kotesovec, Jul 31 2025
a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(2*n+k-1,k). - Seiichi Manyama, Aug 01 2025

A386723 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / (1+2*x)^4 ).

Original entry on oeis.org

1, 11, 175, 3275, 67156, 1460237, 33073930, 771961835, 18437940220, 448483875596, 11071403236807, 276675755470349, 6985664542196380, 177932236341440270, 4566561255466298500, 117974930924420353835, 3065563791639454312492, 80069021664742889373380
Offset: 0

Views

Author

Seiichi Manyama, Jul 31 2025

Keywords

Links

Crossrefs

Cf. A371391, A386722.
Cf. A386719.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-x)^3/(1+2*x)^4)/x)
  • PARI
    a(n) = sum(k=0, n, 2^k*binomial(4*(n+1), k)*binomial(4*n-k+2, n-k))/(n+1);

Formula

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

A386773 Expansion of (1/x) * Series_Reversion( x * (1-3*x)^2 / (1+2*x)^3 ).

Original entry on oeis.org

1, 12, 219, 4778, 115011, 2945982, 78764484, 2172877458, 61393035171, 1767592420152, 51672186100899, 1529632003964688, 45760966837725556, 1381338453812353272, 42020564167060633464, 1286902432396816483218, 39645674268979326240291, 1227773019572168363776092
Offset: 0

Views

Author

Seiichi Manyama, Aug 02 2025

Keywords

Links

Crossrefs

Cf. A386722.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-3*x)^2/(1+2*x)^3)/x)
  • PARI
    a(n) = sum(k=0, n, 2^k*3^(n-k)*binomial(3*(n+1), k)*binomial(3*n-k+1, n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} 2^k * 3^(n-k) * binomial(3*(n+1),k) * binomial(3*n-k+1,n-k).
a(n) = (1/(n+1)) * [x^n] ( (1+2*x)^3 / (1-3*x)^2 )^(n+1).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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