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.

A371742 a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-k,n-2*k).

Original entry on oeis.org

1, 3, 16, 92, 551, 3380, 21065, 132771, 843944, 5399802, 34731776, 224361283, 1454557294, 9458829681, 61670895633, 403003997300, 2638776935215, 17308508054848, 113709379928689, 748069400432262, 4927608724973776, 32495826854732633, 214521754579553129
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2024

Keywords

Crossrefs

Cf. A108081, A371743, A371744.
Cf. A066380, A371754.
Cf. A183160.

Programs

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

Formula

a(n) = [x^n] 1/((1-x-x^2) * (1-x)^(2*n)).
a(n) ~ 3^(3*n + 3/2) / (5 * sqrt(Pi*n) * 2^(2*n)). - Vaclav Kotesovec, Apr 05 2024

A371756 a(n) = Sum_{k=0..floor(n/3)} binomial(5*n-2*k,n-3*k).

Original entry on oeis.org

1, 5, 45, 456, 4863, 53383, 597052, 6765471, 77407257, 892270250, 10346070471, 120542238796, 1410040212166, 16549315766244, 194792566133507, 2298472850258746, 27179673132135409, 322013956853586970, 3821532498419234994, 45420775578132979989
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2024

Keywords

Crossrefs

Cf. A144904, A371754, A371755.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(5*n-2*k, n-3*k));

Formula

a(n) = [x^n] 1/((1-x-x^3) * (1-x)^(4*n)).
a(n) ~ 5^(5*n + 5/2) / (99 * sqrt(Pi*n) * 2^(8*n - 1/2)). - Vaclav Kotesovec, Apr 05 2024
a(n) = binomial(5*n, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [(1-5*n)/2, -5*n/2, 1+4*n], -27/4). - Stefano Spezia, Apr 06 2024

A371774 a(n) = Sum_{k=0..floor(n/3)} binomial(3*n-k+1,n-3*k).

Original entry on oeis.org

1, 4, 21, 121, 727, 4473, 27949, 176549, 1124332, 7205511, 46411744, 300183757, 1948255421, 12681654613, 82755728730, 541213820732, 3546268982757, 23276100962571, 153004515241866, 1007131032951572, 6637396253259291, 43791520333601111
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2024

Keywords

Crossrefs

Cf. A371773, A371775, A371776.
Cf. A371754, A371770.
Cf. A045721.

Programs

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

Formula

a(n) = [x^n] 1/(((1-x)^2-x^3) * (1-x)^(2*n)).
a(n) = binomial(1+3*n, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [-1-3*n, 1+n, 3/2+n], 27/4). - Stefano Spezia, Apr 06 2024
From Vaclav Kotesovec, Apr 08 2024: (Start)
Recurrence: 2*n*(2*n - 1)*(671*n^4 - 4757*n^3 + 11743*n^2 - 11533*n + 3516)*a(n) = (44957*n^6 - 350256*n^5 + 997889*n^4 - 1236792*n^3 + 563834*n^2 + 39768*n - 60480)*a(n-1) - 10*(19459*n^6 - 156741*n^5 + 461272*n^4 - 575421*n^3 + 211099*n^2 + 106572*n - 60480)*a(n-2) + (93269*n^6 - 753150*n^5 + 2221631*n^4 - 2772678*n^3 + 999800*n^2 + 543408*n - 302400)*a(n-3) - 3*(3*n - 8)*(3*n - 7)*(671*n^4 - 2073*n^3 + 1498*n^2 + 366*n - 360)*a(n-4).
a(n) ~ 3^(3*n + 5/2) / (11 * sqrt(Pi*n) * 2^(2*n)). (End)

A371755 a(n) = Sum_{k=0..floor(n/3)} binomial(4*n-2*k,n-3*k).

Original entry on oeis.org

1, 4, 28, 221, 1834, 15657, 136137, 1199014, 10661184, 95493145, 860339723, 7788028028, 70777321331, 645359630071, 5901209474518, 54093485799726, 496910913391428, 4573312196055502, 42160889572810597, 389258294230352460, 3598732127428879981
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2024

Keywords

Crossrefs

Cf. A144904, A371754, A371756.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(4*n-2*k, n-3*k));

Formula

a(n) = [x^n] 1/((1-x-x^3) * (1-x)^(3*n)).
a(n) ~ 2^(8*n + 9/2) / (47 * sqrt(Pi*n) * 3^(3*n - 1/2)). - Vaclav Kotesovec, Apr 05 2024
Showing 1-4 of 4 results.

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