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.

A382031 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x)^2)), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 3, 43, 1177, 46681, 2419291, 154587427, 11735209585, 1031418915121, 102979800567091, 11510663862332251, 1423811747933017609, 193073662118499898633, 28479005472094048953355, 4539456019668776334683731, 777538096585429376795405281, 142419954152382631361835929185
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Programs

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

Formula

Let F(x) be the e.g.f. of A382044. F(x) = log(A(x))/x = B(x*A(x)^2).
a(n) = n! * Sum_{k=0..n-1} (2*k+1)^(n-k-1) * binomial(n+3*k,k)/((n+3*k) * (n-k-1)!) for n > 0.

A382030 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x)^2)), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 3, 37, 817, 25741, 1053211, 52957297, 3157457185, 217695187801, 17036331544531, 1491702434847901, 144479729938558609, 15335923797225215653, 1770255543485671432555, 220776904683577075549801, 29582947262972619472787521, 4238424613351537181204589745, 646565304924896452410832170787
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Programs

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

Formula

Let F(x) be the e.g.f. of A382043. F(x) = log(A(x))/x = B(x*A(x)^2).
a(n) = n! * Sum_{k=0..n-1} (2*k+1)^(n-k-1) * binomial(n+2*k,k)/((n+2*k) * (n-k-1)!) for n > 0.

A382042 E.g.f. A(x) satisfies A(x) = exp(x*C(x*A(x)^3)), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Original entry on oeis.org

1, 1, 3, 37, 733, 20181, 714541, 30903769, 1579206441, 93099946249, 6219777779641, 464382363698661, 38319628830696973, 3463058939163189133, 340172205752538636933, 36087128101110502864561, 4111807211977470782285521, 500807663307856030823859729, 64931674940413564774656214513
Offset: 0

Views

Author

Seiichi Manyama, Mar 13 2025

Keywords

Crossrefs

Programs

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

Formula

Let F(x) be the e.g.f. of A382039. F(x) = log(A(x))/x = C(x*A(x)^3).
E.g.f.: A(x) = exp( Series_Reversion( x*(1 - x*exp(3*x)) ) ).
a(n) = n! * Sum_{k=0..n-1} (3*k+1)^(n-k-1) * binomial(n+k,k)/((n+k) * (n-k-1)!) for n > 0.
Showing 1-3 of 3 results.