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 1-5 of 5 results.

A370101 a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(4*n-k-1,n-k).

Original entry on oeis.org

1, 7, 97, 1519, 25089, 427007, 7408897, 130287871, 2313945089, 41409732607, 745530884097, 13488086405119, 245014271688705, 4465915098890239, 81637668328243201, 1496095489290731519, 27477504726883368961, 505627095685486608383, 9320167322334416338945
Offset: 0

Views

Author

Seiichi Manyama, Feb 10 2024

Keywords

Crossrefs

Cf. A001448, A370100, A370102.
Cf. A119259, A370097.
Cf. A365846.

Programs

  • Mathematica
    Table[Sum[Binomial[4n,k]Binomial[4n-k-1,n-k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Dec 09 2024 *)
Table[Sum[2^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, binomial(4*n, k)*binomial(4*n-k-1, n-k));

Formula

a(n) = [x^n] ( (1+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+x)^4 ). See A365846.
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n,k). - Seiichi Manyama, Jul 31 2025
a(n) ~ 2^(9*n + 3/2) / (7 * sqrt(Pi*n) * 3^(3*n - 1/2)). - Vaclav Kotesovec, Jul 31 2025
a(n) = Sum_{k=0..n} 2^k * binomial(3*n+k-1,k). - Seiichi Manyama, Aug 01 2025
a(n) = [x^n] 1/((1-x) * (1-2*x)^(3*n)). - Seiichi Manyama, Aug 09 2025

A383326 a(n) = Sum_{k=0..n} binomial(3*n+1,k) * binomial(3*n-k,n-k).

Original entry on oeis.org

1, 7, 71, 799, 9439, 114687, 1419263, 17791487, 225172991, 2870945791, 36819740671, 474470776831, 6138443497471, 79681448443903, 1037278449106943, 13536444411412479, 177030837540093951, 2319618918724403199, 30444928900076666879, 400189735705069486079, 5267487129636270243839
Offset: 0

Views

Author

Seiichi Manyama, Aug 04 2025

Keywords

Crossrefs

Cf. A178792, A383716.
Cf. A370097.

Programs

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

Formula

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

A370098 a(n) = Sum_{k=0..n} binomial(3*n,k) * binomial(4*n-k-1,n-k).

Original entry on oeis.org

1, 6, 72, 978, 14016, 207006, 3116952, 47568618, 733189632, 11387193846, 177923724072, 2793666465090, 44042615547456, 696708049377294, 11053262513080440, 175800225426741978, 2802193910116429824, 44752001810800994022, 715924864099841086728
Offset: 0

Views

Author

Seiichi Manyama, Feb 10 2024

Keywords

Crossrefs

Cf. A091527, A370097.
Cf. A370099, A370102.
Cf. A365843.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(3*n, k)*binomial(4*n-k-1, n-k));

Formula

a(n) = [x^n] ( (1+x)^3/(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+x)^3 ). See A365843.
From Seiichi Manyama, Aug 09 2025: (Start)
a(n) = [x^n] (1-x)^(n-1)/(1-2*x)^(3*n).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n,k) * binomial(n-1,n-k).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+k-1,k) * binomial(n-1,n-k). (End)

A386836 a(n) = Sum_{k=0..n} binomial(3*n+2,k) * binomial(3*n-k-1,n-k).

Original entry on oeis.org

1, 7, 70, 782, 9199, 111465, 1376764, 17234600, 217891693, 2775766091, 35574777154, 458169648722, 5924747347835, 76876586813629, 1000418599504408, 13051488907037580, 170643358430006553, 2235400439909584575, 29333436132847784062, 385507257723471794774, 5073372058467119928391
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2025

Keywords

Crossrefs

Cf. A386835, A386837.
Cf. A370097, A386833.

Programs

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

Formula

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

A386833 a(n) = Sum_{k=0..n} binomial(3*n+1,k) * binomial(3*n-k-1,n-k).

Original entry on oeis.org

1, 6, 59, 656, 7701, 93210, 1150495, 14395428, 181936169, 2317140014, 29691138099, 382334271544, 4943464235069, 64137141682242, 834561532624967, 10886878474010700, 142332442919829585, 1864423992564121686, 24464149489904517211, 321499324010641490016, 4230840338116037836901
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2025

Keywords

Crossrefs

Cf. A116881, A386834.
Cf. A370097, A386836.
Cf. A383326.

Programs

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

Formula

a(n) = [x^n] (1+x)^(3*n+1)/(1-x)^(2*n).
a(n) = [x^n] 1/((1-x)^2 * (1-2*x)^(2*n)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * (n-k+1) * binomial(3*n+1,k).
a(n) = Sum_{k=0..n} 2^k * (n-k+1) * binomial(2*n+k-1,k).
Showing 1-5 of 5 results.

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

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