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.

Previous Showing 21-27 of 27 results.

A379285 G.f. A(x) satisfies A(x) = 1/((1 - x*A(x)^2) * (1 - x*A(x)^3)).

Original entry on oeis.org

1, 2, 13, 115, 1175, 13052, 153115, 1866599, 23414063, 300238945, 3917984904, 51862207151, 694670871393, 9398137507922, 128235826442635, 1762706644013297, 24386388751113511, 339295523459625535, 4744546261930628062, 66644485202547680010, 939916204595095866644
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2024

Keywords

Crossrefs

Cf. A118969, A199475, A379209, A379284, A379286, A379287.

Programs

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

Formula

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

A379286 G.f. A(x) satisfies A(x) = 1/((1 - x*A(x)^2) * (1 - x*A(x)^5)).

Original entry on oeis.org

1, 2, 17, 209, 3025, 47975, 806673, 14126236, 254880645, 4705443504, 88458542000, 1687588704861, 32589587581341, 635824437818621, 12513756861585915, 248148065577971460, 4953215882123744005, 99442753396113435246, 2006704742456528041800, 40679834776076235917841
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2024

Keywords

Crossrefs

Cf. A118969, A199475, A379209, A379284, A379285, A379287.

Programs

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

Formula

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

A364589 G.f. satisfies A(x) = 1/(1-x) + x^3*A(x)^3.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 14, 31, 67, 146, 331, 760, 1749, 4072, 9583, 22673, 53929, 129055, 310328, 749152, 1815481, 4415313, 10771564, 26352955, 64644926, 158963191, 391767016, 967523138, 2394060433, 5934576763, 14735792889, 36647185192, 91274339014, 227645446307
Offset: 0

Views

Author

Seiichi Manyama, Jul 29 2023

Keywords

Crossrefs

Cf. A216604, A226974.
Cf. A049130, A199475, A364590.

Programs

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

Formula

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

A364590 G.f. satisfies A(x) = 1/(1-x) + x^4*A(x)^3.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 7, 11, 19, 37, 74, 142, 268, 518, 1033, 2077, 4152, 8290, 16687, 33899, 69148, 141160, 288650, 592354, 1220086, 2519226, 5210164, 10794088, 22408556, 46613554, 97125751, 202662419, 423459427, 886048249, 1856448852, 3894362560, 8178530890
Offset: 0

Views

Author

Seiichi Manyama, Jul 29 2023

Keywords

Crossrefs

Cf. A346073, A364591.
Cf. A049130, A199475, A364589.

Programs

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

Formula

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

A366178 G.f. A(x) satisfies A(x) = 1/(1 - x) + x*A(x)^3/(1 - x)^3.

Original entry on oeis.org

1, 2, 10, 67, 502, 4045, 34279, 301232, 2720266, 25091431, 235394601, 2239139980, 21546299491, 209361514219, 2051379996574, 20245794958408, 201079938971546, 2008276118393320, 20157131084034349, 203215717750220949, 2056913539436637829
Offset: 0

Views

Author

Seiichi Manyama, Oct 03 2023

Keywords

Crossrefs

Partial sums give A366179.
Cf. A199475, A346626, A366177.

Programs

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

Formula

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

A379281 G.f. A(x) satisfies A(x) = 1/( (1 - x) * (1 - x*A(x)) )^2.

Original entry on oeis.org

1, 4, 18, 96, 575, 3706, 25078, 175666, 1262723, 9261018, 69024147, 521281642, 3980391050, 30678331440, 238350850248, 1864751821958, 14678131286357, 116160233811868, 923684828888152, 7376541052964806, 59137050311947284, 475757909357776656, 3839678158239147611
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2024

Keywords

Crossrefs

Cf. A379282, A379283.
Cf. A199475.

Programs

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

Formula

G.f.: B(x)^2 where B(x) is the g.f. of A199475.
a(n) = 2 * Sum_{k=0..n} binomial(2*n-k+2,k) * binomial(3*n-3*k+1,n-k)/(2*n-k+2).

A366177 G.f. A(x) satisfies A(x) = 1/(1 - x) + x*A(x)^3/(1 - x)^2.

Original entry on oeis.org

1, 2, 9, 55, 382, 2866, 22648, 185722, 1565725, 13486036, 118163960, 1049908872, 9437623630, 85671158757, 784247925911, 7231502249005, 67106161264660, 626221543735984, 5872908642398977, 55323451127462123, 523240983692525619, 4966658879361416551
Offset: 0

Views

Author

Seiichi Manyama, Oct 03 2023

Keywords

Crossrefs

Partial sums give A364620.
Cf. A199475, A346626, A366178.

Programs

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

Formula

a(n) = Sum_{k=0..n} binomial(n+3*k,n-k) * binomial(3*k,k)/(2*k+1).
Previous Showing 21-27 of 27 results.

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