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.

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A379087 a(n) = Sum_{k=0..floor(n/3)} binomial(3*n+k-1,k) * binomial(3*n+k,n-3*k).

Original entry on oeis.org

1, 3, 15, 93, 651, 4803, 36177, 275208, 2108091, 16243671, 125817345, 978933354, 7646000421, 59915086026, 470820659940, 3708756501018, 29276677544619, 231540519752376, 1834228504348863, 14552075416977531, 115605043235217081, 919503729585453147
Offset: 0

Views

Author

Seiichi Manyama, Dec 15 2024

Keywords

Crossrefs

Cf. A379025, A379086.
Cf. A379090.

Programs

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

Formula

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

A379083 Expansion of (1/x) * Series_Reversion( x * (1/(1 + x) - x^3)^3 ).

Original entry on oeis.org

1, 3, 12, 58, 321, 1941, 12405, 82188, 558567, 3870694, 27245268, 194269872, 1400352702, 10187886330, 74710928103, 551676261727, 4098401671788, 30610414484517, 229717037309281, 1731295701244008, 13098454442320593, 99444838611953627, 757393732018935552, 5785220154325055826
Offset: 0

Views

Author

Seiichi Manyama, Dec 15 2024

Keywords

Crossrefs

Cf. A198888, A379082.
Cf. A379087, A379090.

Programs

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

Formula

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{k>=1} A379087(k) * x^k/k ).
(2) A(x) = ( (1 + x*A(x)) * (1 + x^3*A(x)^(10/3)) )^3.
(3) A(x) = B(x)^3 where B(x) is the g.f. of A379090.
a(n) = (1/(n+1)) * [x^n] 1/( 1/(1 + x) - x^3 )^(3*(n+1)).
a(n) = 3 * Sum_{k=0..floor(n/3)} binomial(3*n+k+3,k) * binomial(3*n+k+3,n-3*k)/(3*n+k+3) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(3*n+k+2,k) * binomial(3*n+k+3,n-3*k).
Showing 1-2 of 2 results.

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