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.

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

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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).

A381913 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / B(x) ), where B(x) is the g.f. of A001764.

Original entry on oeis.org

1, 4, 28, 245, 2422, 25860, 291106, 3405405, 41014131, 505344113, 6341182427, 80768735045, 1041645452650, 13575670575944, 178528253213469, 2366073408348545, 31571528771106126, 423794981085407622, 5718929869862880055, 77539914280883389432, 1055790501909183080512
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(4*n-k+2, n-k)/(n+3*k+1));

Formula

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

A381914 Expansion of (1/x) * Series_Reversion( x * (1-x) / B(x) ), where B(x) is the g.f. of A002293.

Original entry on oeis.org

1, 2, 10, 72, 624, 6009, 61809, 664813, 7384613, 84045565, 974913510, 11483316680, 136974177209, 1651166320547, 20083352214058, 246168280262403, 3037682020219285, 37706043912831337, 470482875049515074, 5897864081341146065, 74243055437832292562, 938101296155866961124
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(2*n-k, n-k)/(n+4*k+1));

Formula

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

A381905 Expansion of (1/x) * Series_Reversion( x / ((1+x) * B(x)) ), where B(x) is the g.f. of A001764.

Original entry on oeis.org

1, 2, 8, 47, 331, 2570, 21204, 182383, 1617163, 14675783, 135643839, 1272434069, 12083390801, 115934171020, 1122129142754, 10943574296787, 107433077283767, 1060800046515405, 10528321010319417, 104972259713887665, 1050936451974803973, 10560662821468607719
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(n+1, n-k)/(n+3*k+1));

Formula

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