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 1-2 of 2 results.

A379086 a(n) = Sum_{k=0..floor(n/2)} binomial(3*n+k-1,k) * binomial(3*n+k,n-2*k).

Original entry on oeis.org

1, 3, 21, 174, 1509, 13443, 121962, 1120899, 10401021, 97230090, 914283621, 8638552464, 81945757734, 779949538176, 7444735446813, 71237074583589, 683125330952205, 6563268117869076, 63164380112090814, 608805362150884731, 5875874727915635409, 56780302474503539427, 549294315060885105744
Offset: 0

Views

Author

Seiichi Manyama, Dec 15 2024

Keywords

Crossrefs

Cf. A379025, A379087.
Cf. A379088.

Programs

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

Formula

a(n) = [x^n] 1/( 1/(1 + x) - x^2 )^(3*n).
a(n) == 0 (mod 3) for n>0.

A379081 Expansion of (1/x) * Series_Reversion( x * (1/(1 + x) - x^2)^3 ).

Original entry on oeis.org

1, 3, 15, 94, 657, 4905, 38299, 308928, 2554092, 21528728, 184318944, 1598427531, 14011401996, 123946608699, 1105090991634, 9920335032821, 89589290332200, 813367589142888, 7419376746340780, 67965042988027335, 624971955439306953, 5766825797557702751, 53380176096582823851
Offset: 0

Views

Author

Seiichi Manyama, Dec 15 2024

Keywords

Crossrefs

Cf. A181734, A379080.
Cf. A379086, A379088.

Programs

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

Formula

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{k>=1} A379086(k) * x^k/k ).
(2) A(x) = ( (1 + x*A(x)) * (1 + x^2*A(x)^(7/3)) )^3.
(3) A(x) = B(x)^3 where B(x) is the g.f. of A379088.
a(n) = (1/(n+1)) * [x^n] 1/( 1/(1 + x) - x^2 )^(3*(n+1)).
a(n) = 3 * Sum_{k=0..floor(n/2)} binomial(3*n+k+3,k) * binomial(3*n+k+3,n-2*k)/(3*n+k+3) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(3*n+k+2,k) * binomial(3*n+k+3,n-2*k).
Showing 1-2 of 2 results.

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