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.

A381879 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / C(x) ), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 3, 16, 106, 788, 6292, 52743, 457946, 4083328, 37174786, 344142192, 3229827900, 30661272627, 293907951057, 2840826401664, 27657352868946, 270968414904700, 2669604470832568, 26431802684789970, 262864480970961882, 2624640191306617088, 26301183967687772360
Offset: 0

Views

Author

Seiichi Manyama, Mar 09 2025

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2*2*x/(1-sqrt(1-4*x)))/x)

Formula

G.f. A(x) satisfies A(x) = C(x*A(x)) / (1 - x*A(x))^2.
a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(3*n-k+1,n-k)/(n+2*k+1).

A381882 Expansion of (1/x) * Series_Reversion( x / ((1+x)^3 * C(x)) ), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 4, 24, 175, 1428, 12525, 115468, 1103777, 10844715, 108860766, 1111722956, 11514401451, 120666441067, 1277161022725, 13633269293868, 146606818816257, 1586739194404521, 17271207134469417, 188942438655850740, 2076317084779878706, 22909617070555385010
Offset: 0

Views

Author

Seiichi Manyama, Mar 09 2025

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^3*(1-sqrt(1-4*x))/(2*x)))/x)

Formula

G.f. A(x) satisfies A(x) = (1 + x*A(x))^3 * C(x*A(x)).
a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(3*n+3,n-k)/(n+2*k+1).
a(n) = binomial(3*(1 + n), n)*hypergeom([(1+n)/2, 1+n/2, -n], [2 + n, 4 + 2*n], -4)/(1 + n). - Stefano Spezia, Mar 09 2025

A381906 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * B(x)) ), where B(x) is the g.f. of A001764.

Original entry on oeis.org

1, 3, 15, 100, 787, 6848, 63583, 617350, 6191888, 63650430, 667043379, 7099806346, 76538663840, 833975952491, 9169925032189, 101616966476850, 1133736002540882, 12724529836447420, 143567856744995568, 1627454706916166076, 18526192807286106198, 211694470334287787868
Offset: 0

Views

Author

Seiichi Manyama, Mar 10 2025

Keywords

Crossrefs

Programs

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

Formula

G.f. A(x) satisfies A(x) = (1 + x*A(x))^2 * B(x*A(x)).
a(n) = Sum_{k=0..n} binomial(n+3*k+1,k) * binomial(2*n+2,n-k)/(n+3*k+1).
Showing 1-3 of 3 results.