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 1-4 of 4 results.

A383888 a(n) = Sum_{k=0..n} 3^k * binomial(n+k-1,k).

Original entry on oeis.org

1, 4, 34, 334, 3478, 37384, 409960, 4558306, 51199558, 579554056, 6600532684, 75546800476, 868224027916, 10012494936136, 115804853315332, 1342795688895754, 15604522381828678, 181690692393744376, 2119144763079629452, 24754486729805925124, 289563977079418497748
Offset: 0

Views

Author

Seiichi Manyama, Aug 01 2025

Keywords

Links

Crossrefs

Cf. A384950, A385438.
Cf. A064063, A385319.

Programs

  • PARI
    a(n) = sum(k=0, n, 3^k*binomial(n+k-1,k));

Formula

a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n,k) * binomial(2*n-k-1,n-k).
a(n) = [x^n] ( (1+x)^2/(1-2*x) )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-2*x) / (1+x)^2 ).
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(2*n,k).
a(n) = (-2)^(-n)*(1 - (-6)^n*binomial(2*n-1, n)*(hypergeom([1, 2*n], [1+n], 3) - 1)). - Stefano Spezia, Aug 02 2025
a(n) ~ 2^(2*n) * 3^(n+1) / (5*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 04 2025

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

A386719 a(n) = Sum_{k=0..n} 2^k * binomial(4*n,k) * binomial(4*n-k-1,n-k).

Original entry on oeis.org

1, 11, 229, 5381, 133333, 3404156, 88600483, 2337160718, 62263902037, 1671407550260, 45137852641204, 1224954657942125, 33377579214681619, 912572183952374996, 25023054179816358034, 687862647149533181036, 18950129471489195622229, 523067259899842250453060
Offset: 0

Views

Author

Seiichi Manyama, Jul 31 2025

Keywords

Crossrefs

Cf. A385319, A385320.
Cf. A386723.

Programs

  • Mathematica
    Table[Sum[3^k*(-1)^(n-k)*Binomial[4*n, k], {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jul 31 2025 *)
  • PARI
    a(n) = sum(k=0, n, 2^k*binomial(4*n, k)*binomial(4*n-k-1, n-k));

Formula

a(n) = [x^n] ( (1+2*x)^4/(1-x)^3 )^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)^3 / (1+2*x)^4 ). See A386723.
a(n) = Sum_{k=0..n} 3^k * (-1)^(n-k) * binomial(4*n,k).
a(n) ~ 2^(8*n - 1/2) / (5 * sqrt(Pi*n) * 3^(2*n - 3/2)). - Vaclav Kotesovec, Jul 31 2025
a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(3*n+k-1,k). - Seiichi Manyama, Aug 01 2025

A383884 a(n) = Sum_{k=0..n} 2^k * binomial(2*n+1,k) * binomial(2*n-k,n-k).

Original entry on oeis.org

1, 8, 76, 776, 8236, 89528, 989080, 11055248, 124659148, 1415338328, 16157960776, 185298481904, 2133004809976, 24631812347696, 285225658980016, 3310631101181216, 38506555289077516, 448698354100917656, 5236993294930652776, 61212903131657378096, 716430640316516361256
Offset: 0

Views

Author

Seiichi Manyama, Aug 04 2025

Keywords

Crossrefs

Cf. A385319.

Programs

  • PARI
    a(n) = sum(k=0, n, 2^k*binomial(2*n+1, k)*binomial(2*n-k, n-k));

Formula

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

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