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.

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.

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.

A382044 E.g.f. A(x) satisfies A(x) = 1 + x*A(x)^4*exp(2*x*A(x)).

Original entry on oeis.org

1, 1, 12, 252, 8096, 352120, 19372512, 1290832480, 101078857728, 9098805892608, 925857411706880, 105098610198360064, 13167689873652178944, 1804954814456584081408, 268702350796640969736192, 43172786067215188056023040, 7446421094705349321120677888, 1372319952106065844255081037824
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+3*k+1, k)/((n+3*k+1)*(n-k)!));

Formula

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

A382040 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*exp(4*x)) ).

Original entry on oeis.org

1, 1, 12, 198, 4912, 163120, 6796224, 341366704, 20088997632, 1356164492544, 103333898644480, 8773563043734016, 821474949840482304, 84093840447771701248, 9344359942839980900352, 1120159940123276849141760, 144096985208727744665288704, 19800296439825918648654561280
Offset: 0

Views

Author

Seiichi Manyama, Mar 13 2025

Keywords

Crossrefs

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = 1 + x*A(x)^2*exp(4*x*A(x)).
a(n) = (1/(n+1)) * Sum_{k=0..n} (4*k)^(n-k) * (n+k)!/(k! * (n-k)!).
Showing 1-4 of 4 results.