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.

A386763 a(n) = Sum_{k=0..n} 5^k * 3^(n-k) * binomial(n+k-1,k).

Original entry on oeis.org

1, 8, 114, 1862, 32246, 576768, 10529544, 194960802, 3647285766, 68772760928, 1304858513324, 24882531221292, 476462691535436, 9155397868559288, 176447193966483204, 3409285356643013082, 66020593061854488006, 1280989373915746600848, 24897996624141835608684
Offset: 0

Views

Author

Seiichi Manyama, Aug 02 2025

Keywords

Crossrefs

Cf. A386764, A386765.
Cf. A383888, A386769.

Programs

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

Formula

a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(2*n,k) * binomial(2*n-k-1,n-k).
a(n) = [x^n] ( (1+3*x)^2/(1-2*x) )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-2*x) / (1+3*x)^2 ). See A386769.
a(n) = Sum_{k=0..n} 5^k * (-2)^(n-k) * binomial(2*n,k).
a(n) = (-9/2)^n*(1 - (-10/9)^n*binomial(2*n-1, n)*(hypergeom([1, 2*n], [1+n], 5/3) - 1)). - Stefano Spezia, Aug 02 2025

A386764 a(n) = Sum_{k=0..n} 5^k * 3^(n-k) * binomial(2*n+k-1,k).

Original entry on oeis.org

1, 13, 319, 8872, 260511, 7885793, 243404884, 7615561092, 240662849871, 7663737420223, 245529092332599, 7904950462600512, 255541233005365956, 8289112264436610828, 269663237466343607464, 8794852773491081069132, 287467221911677590185391, 9414259968096351504747483
Offset: 0

Views

Author

Seiichi Manyama, Aug 02 2025

Keywords

Crossrefs

Cf. A386763, A386765.
Cf. A384950, A386770.

Programs

  • PARI
    a(n) = sum(k=0, n, 5^k*3^(n-k)*binomial(2*n+k-1, k));

Formula

a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k).
a(n) = [x^n] ( (1+3*x)^3/(1-2*x)^2 )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-2*x)^2 / (1+3*x)^3 ). See A386770.
a(n) = Sum_{k=0..n} 5^k * (-2)^(n-k) * binomial(3*n,k).
a(n) = (27/4)^n - 5^n*binomial(3*n-1, n)*(hypergeom([1, 3*n], [1+n], 5/3) - 1). - Stefano Spezia, Aug 02 2025

A386831 a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).

Original entry on oeis.org

1, 23, 814, 32102, 1330436, 56734023, 2464566064, 108464237352, 4819668737436, 215760575713148, 9716002818365314, 439628651114930102, 19971546503835844436, 910318041046245082898, 41611957337801849102064, 1906855257451887625497852, 87569968895543824193201436
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2025

Keywords

Crossrefs

Cf. A386829, A386830.
Cf. A386765.

Programs

  • PARI
    a(n) = sum(k=0, n, 3^k*2^(n-k)*binomial(4*n+1, k)*binomial(4*n-k, n-k));

Formula

a(n) = [x^n] (1+3*x)^(4*n+1)/(1-2*x)^(3*n+1).
a(n) = [x^n] 1/((1-3*x) * (1-5*x)^(3*n+1)).
a(n) = Sum_{k=0..n} 5^k * (-2)^(n-k) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} 5^k * 3^(n-k) * binomial(3*n+k,k).
Showing 1-3 of 3 results.

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