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.

A371754 a(n) = Sum_{k=0..floor(n/3)} binomial(3*n-2*k,n-3*k).

Original entry on oeis.org

1, 3, 15, 85, 505, 3081, 19125, 120173, 761995, 4865697, 31244029, 201544551, 1305039209, 8477521051, 55221311565, 360559717807, 2359123470971, 15463951609491, 101530816122729, 667587477393509, 4395294402200983, 28972295880583861, 191181607835416543
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2024

Keywords

Crossrefs

Cf. A144904, A371755, A371756.
Cf. A066380, A371742.

Programs

  • Mathematica
    Table[Sum[Binomial[3n-2k,n-3k],{k,0,Floor[n/3]}],{n,0,30}] (* Harvey P. Dale, Oct 19 2024 *)
  • PARI
    a(n) = sum(k=0, n\3, binomial(3*n-2*k, n-3*k));

Formula

a(n) = [x^n] 1/((1-x-x^3) * (1-x)^(2*n)).
a(n) ~ 3^(3*n + 5/2) / (17 * sqrt(Pi*n) * 2^(2*n)). - Vaclav Kotesovec, Apr 05 2024

A371772 a(n) = Sum_{k=0..floor(n/3)} binomial(5*n-3*k-1,n-3*k).

Original entry on oeis.org

1, 4, 36, 365, 3892, 42714, 477621, 5411109, 61901268, 713435333, 8271470666, 96361329024, 1127086021461, 13227336997645, 155680966681101, 1836862248992565, 21719923705450260, 257316706385394615, 3053599633736172765, 36292098436808314572, 431918050456887676362
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2024

Keywords

Crossrefs

Cf. A371758, A371770, A371771.
Cf. A371756.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(5*n-3*k-1, n-3*k));

Formula

a(n) = [x^n] 1/((1-x^3) * (1-x)^(4*n)).
a(n) = binomial(5*n-1, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [(1-5*n)/3, (2-5*n)/3, 1-5*n/3], 1). - Stefano Spezia, Apr 06 2024
From Vaclav Kotesovec, Apr 08 2024: (Start)
Recurrence: 72*n*(2*n - 1)*(4*n - 3)*(4*n - 1)*(899*n^2 - 2355*n + 1534)*a(n) = (25514519*n^6 - 117751221*n^5 + 212960873*n^4 - 191684487*n^3 + 89835824*n^2 - 20567076*n + 1769040)*a(n-1) - 5*(5*n - 7)*(5*n - 6)*(5*n - 4)*(5*n - 3)*(899*n^2 - 557*n + 78)*a(n-2).
a(n) ~ 5^(5*n + 5/2) / (31 * sqrt(Pi*n) * 2^(8*n + 3/2)). (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 1-3 of 3 results.

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

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