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 1-4 of 4 results.

A381880 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, 27, 223, 2052, 20199, 208205, 2219149, 24261279, 270581313, 3066581130, 35216499786, 408919039968, 4792955710138, 56633333886618, 673881539636365, 8067939162382594, 97117925556632184, 1174721577627568371, 14270877151754826473, 174044527062280321368
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*2*x/(1-sqrt(1-4*x)))/x)

Formula

G.f. A(x) satisfies A(x) = C(x*A(x)) / (1 - x*A(x))^3.
a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(4*n-k+2,n-k)/(n+2*k+1).

A381881 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, 14, 82, 547, 3958, 30249, 240362, 1966235, 16449495, 140093989, 1210575512, 10587490383, 93540456103, 833619150838, 7484887130882, 67645312129491, 614872423359187, 5617522739173495, 51556112664387720, 475105557839611760, 4394434006611790855
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*(1-sqrt(1-4*x))/(2*x)))/x)

Formula

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

A381912 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, 17, 124, 1038, 9470, 91586, 923542, 9608323, 102403921, 1112500651, 12275235274, 137193964646, 1549964417407, 17672282336488, 203092563108610, 2350061579393077, 27357919380212638, 320186582453226290, 3765185566095185740, 44465070300433434901, 527131055014319691537
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(3*n-k+1, n-k)/(n+3*k+1));

Formula

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

A381915 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / B(x) ), where B(x) is the g.f. of A002293.

Original entry on oeis.org

1, 3, 18, 145, 1378, 14515, 163700, 1936414, 23716654, 298216851, 3827542585, 49938733635, 660366743580, 8830549084588, 119205253249287, 1622258295003714, 22232669093660250, 306569446979862205, 4250285556933578693, 59210418891925845529, 828417259759216617257
Offset: 0

Views

Author

Seiichi Manyama, Mar 10 2025

Keywords

Crossrefs

Programs

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

Formula

G.f. A(x) satisfies A(x) = B(x*A(x)) / (1 - x*A(x))^2.
a(n) = Sum_{k=0..n} binomial(n+4*k+1,k) * binomial(3*n-k+1,n-k)/(n+4*k+1).
Showing 1-4 of 4 results.