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.

A365856 Expansion of (1/x) * Series_Reversion( x*(1+x)^2*(1-x)^5 ).

Original entry on oeis.org

1, 3, 17, 115, 863, 6903, 57687, 497683, 4398980, 39630305, 362562226, 3359252039, 31457036708, 297247495745, 2830725974514, 27140465365203, 261768686779800, 2538061348959000, 24724191398850125, 241862002342417585, 2374978445599884762
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Links

Crossrefs

Cf. A365854, A365855.
Cf. A365753.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^k*binomial(2*n+k+1, k)*binomial(6*n-k+4, n-k))/(n+1);
  • SageMath
    def A365856(n):
    h = binomial(6*n + 4, n) * hypergeometric([-n, 2*n + 2], [-6 * n - 4], -1) / (n + 1)
    return simplify(h)
print([A365856(n) for n in range(21)])  # Peter Luschny, Sep 20 2023

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(2*n+k+1,k) * binomial(6*n-k+4,n-k).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+k+1,k) * binomial(4*n-2*k+2,n-2*k). - Seiichi Manyama, Jan 18 2024
a(n) = (1/(n+1)) * [x^n] 1/( (1+x)^2 * (1-x)^5 )^(n+1). - Seiichi Manyama, Feb 16 2024

A365855 Expansion of (1/x) * Series_Reversion( x*(1+x)^2*(1-x)^4 ).

Original entry on oeis.org

1, 2, 9, 46, 264, 1612, 10291, 67830, 458109, 3153744, 22049065, 156127140, 1117369884, 8069610992, 58735003740, 430416574918, 3172987081311, 23514565653058, 175083678670264, 1309132916709168, 9825882638364144, 74003924059921940, 559112987425763365
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Links

Crossrefs

Cf. A365854, A365856.
Cf. A365752, A368967.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^k*binomial(2*n+k+1, k)*binomial(5*n-k+3, n-k))/(n+1);
  • SageMath
    def A365855(n):
    h = binomial(5*n + 3, n) * hypergeometric([-n, 2*n + 2], [-5 * n - 3], -1) / (n + 1)
    return simplify(h)
print([A365855(n) for n in range(23)])  # Peter Luschny, Sep 20 2023

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(2*n+k+1,k) * binomial(5*n-k+3,n-k).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+k+1,k) * binomial(3*n-2*k+1,n-2*k). - Seiichi Manyama, Jan 18 2024
a(n) = (1/(n+1)) * [x^n] 1/( (1+x)^2 * (1-x)^4 )^(n+1). - Seiichi Manyama, Feb 16 2024

A370103 a(n) = Sum_{k=0..n} (-1)^k * binomial(2*n+k-1,k) * binomial(4*n-k-1,n-k).

Original entry on oeis.org

1, 1, 7, 28, 151, 751, 3976, 20924, 112023, 602182, 3260257, 17724928, 96766072, 529977917, 2910984412, 16027963528, 88440034711, 488918693466, 2707393587802, 15014647096172, 83380131228401, 463593653171495, 2580426581343200, 14377474236172320
Offset: 0

Views

Author

Seiichi Manyama, Feb 10 2024

Keywords

Crossrefs

Cf. A351857, A370104.
Cf. A352373, A370098, A370101.
Cf. A001700, A348410.
Cf. A365854.

Programs

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

Formula

a(n) = [x^n] 1/( (1+x)^2 * (1-x)^3 )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x*(1+x)^2*(1-x)^3 ). See A365854.
a(n) = Sum_{k=0..floor(n/2)} binomial(3*n+k-1,k) * binomial(n,n-2*k).
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n+k-1,k) * binomial(2*n-2*k-1,n-2*k).
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.