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.

A379085 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n+k-1,k) * binomial(2*n+k,n-3*k).

Original entry on oeis.org

1, 2, 6, 26, 142, 802, 4434, 24222, 132686, 733076, 4081926, 22853052, 128427106, 723862856, 4090573570, 23170106086, 131515806574, 747875338152, 4259810283828, 24298797944956, 138787172202182, 793651842511512, 4543393775520936, 26035130683198684, 149325002408646002
Offset: 0

Views

Author

Seiichi Manyama, Dec 15 2024

Keywords

Crossrefs

Cf. A379022, A379084.
Cf. A379089.

Programs

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

Formula

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

A379080 Expansion of (1/x) * Series_Reversion( x * (1/(1 + x) - x^2)^2 ).

Original entry on oeis.org

1, 2, 7, 32, 163, 886, 5039, 29616, 178446, 1096356, 6842452, 43259122, 276462247, 1783114592, 11591769207, 75874998822, 499643588823, 3307746965238, 22001986381873, 146972401234478, 985535271867577, 6631547191254298, 44763982636889092, 303037237861086682
Offset: 0

Views

Author

Seiichi Manyama, Dec 15 2024

Keywords

Crossrefs

Cf. A181734, A379081.
Cf. A200719, A379084.

Programs

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

Formula

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{k>=1} A379084(k) * x^k/k ).
(2) A(x) = ( (1 + x*A(x)) * (1 + x^2*A(x)^(5/2)) )^2.
(3) A(x) = B(x)^2 where B(x) is the g.f. of A200719.
a(n) = (1/(n+1)) * [x^n] 1/( 1/(1 + x) - x^2 )^(2*(n+1)).
a(n) = 2 * Sum_{k=0..floor(n/2)} binomial(2*n+k+2,k) * binomial(2*n+k+2,n-2*k)/(2*n+k+2) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+k+1,k) * binomial(2*n+k+2,n-2*k).
Showing 1-2 of 2 results.

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

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