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.

A365726 G.f. satisfies A(x) = 1 + x^3*A(x)^5*(1 + x*A(x)).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 5, 11, 6, 35, 120, 136, 336, 1330, 2310, 4301, 15456, 35100, 64701, 193662, 508921, 1023000, 2643432, 7298984, 16196682, 38795055, 105939288, 254015541, 596987183, 1575487320, 3959803694, 9418896773, 24081344034, 61781452530, 150293865540
Offset: 0

Views

Author

Seiichi Manyama, Sep 17 2023

Keywords

Crossrefs

Cf. A308616, A365723, A365724, A365725.
Cf. A365695.

Programs

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

Formula

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

A365694 G.f. satisfies A(x) = 1 + x^3*A(x)^2 / (1 - x*A(x)).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 3, 6, 10, 20, 42, 84, 170, 354, 740, 1549, 3269, 6945, 14811, 31711, 68177, 147091, 318313, 690837, 1503351, 3279445, 7169907, 15708485, 34482475, 75830981, 167042763, 368548926, 814341362, 1801867812, 3992172298, 8855912464, 19668236110
Offset: 0

Views

Author

Seiichi Manyama, Sep 16 2023

Keywords

Crossrefs

Cf. A023426, A023432, A054514, A114997, A365695.
Cf. A365243.

Programs

  • Mathematica
    CoefficientList[Series[2/(1 + x + Sqrt[1 + x*(-2 + x - 4*x^2)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 16 2023 *)
  • PARI
    a(n) = sum(k=0, n\3, binomial(n-2*k-1, n-3*k)*binomial(n-k+1, k)/(n-k+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k-1,n-3*k) * binomial(n-k+1,k) / (n-k+1).
G.f.: A(x) = 2/(1 + x + sqrt(1 + x*(-2 + x - 4*x^2))). - Vaclav Kotesovec, Sep 16 2023

A365757 G.f. satisfies A(x) = 1 + x*A(x) / (1 - x^3*A(x)^5).

Original entry on oeis.org

1, 1, 1, 1, 2, 8, 29, 86, 229, 619, 1836, 5846, 18802, 59356, 185187, 581476, 1855412, 5997965, 19491730, 63395718, 206433172, 674452128, 2213463944, 7293253791, 24098638133, 79791002807, 264698873350, 879945619711, 2931486913728, 9785457123420, 32721317536787
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2023

Keywords

Crossrefs

Cf. A023432, A212383, A365243, A365756.
Cf. A365695, A365759.

Programs

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

Formula

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

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

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