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.

A382029 E.g.f. A(x) satisfies A(x) = exp(x*C(x*A(x)^2)), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Original entry on oeis.org

1, 1, 3, 31, 529, 12601, 385891, 14440567, 638576065, 32580927505, 1883889232291, 121742057314351, 8695278706372369, 680187946863332233, 57833833258995140803, 5310742450917819399751, 523793286672328763358721, 55223769332070053104438945, 6197871354601209094032190147
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+k, k)/((n+k)*(n-k-1)!)));

Formula

Let F(x) be the e.g.f. of A379690. F(x) = log(A(x))/x = C(x*A(x)^2).
E.g.f.: A(x) = exp( Series_Reversion( x*(1 - x*exp(2*x)) ) ).
a(n) = n! * Sum_{k=0..n-1} (2*k+1)^(n-k-1) * binomial(n+k,k)/((n+k) * (n-k-1)!) for n > 0.

A382043 E.g.f. A(x) satisfies A(x) = 1 + x*A(x)^3*exp(2*x*A(x)).

Original entry on oeis.org

1, 1, 10, 168, 4280, 146840, 6354432, 332467072, 20419261312, 1440559380096, 114820434103040, 10205253450850304, 1000815286620229632, 107355373421379825664, 12504295470535952613376, 1571670041412254073323520, 212035122185327799251468288, 30561822671438790519426154496
Offset: 0

Views

Author

Seiichi Manyama, Mar 13 2025

Keywords

Crossrefs

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (2*k)^(n-k) * binomial(n+2*k+1,k)/((n+2*k+1) * (n-k)!).
Showing 1-3 of 3 results.