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.

A295238 Expansion of e.g.f. 2/(1 + sqrt(1 - 4*x*exp(x))).

Original entry on oeis.org

1, 1, 6, 57, 796, 14785, 344046, 9640225, 316255416, 11896233345, 504918768250, 23874754106401, 1244712973780068, 70940791877082049, 4388291507415513894, 292823509879910802465, 20966854494419642792176, 1603540841320336494905089, 130464295561360336835272050
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 18 2017

Keywords

Comments

Inverse binomial transform of A194471.

Links

Crossrefs

Cf. A000108, A006531, A052895, A194471, A213644, A295239.

Programs

  • Maple
    a:=series(2/(1+sqrt(1-4*x*exp(x))),x=0,19): seq(n!*coeff(a,x,n),n=0..18); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 18; CoefficientList[Series[2/(1 + Sqrt[1 - 4 x Exp[x]]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 18; CoefficientList[Series[1/(1 + ContinuedFractionK[-x Exp[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n, k] k! Sum[(m + 1)^(k - m - 1) Binomial[2 m, m]/(k - m)!, {m, 0, k}], {k, 0, n}], {n, 0, 18}]
  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(2*k, k)/((k+1)*(n-k)!)); \\ Seiichi Manyama, Aug 15 2023

Formula

E.g.f.: 1/(1 - x*exp(x)/(1 - x*exp(x)/(1 - x*exp(x)/(1 - x*exp(x)/(1 - ...))))), a continued fraction.
a(n) ~ sqrt(2*(1 + LambertW(1/4))) * n^(n-1) / ((LambertW(1/4))^n * exp(n)). - Vaclav Kotesovec, Nov 18 2017
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(2*k+1,k)/( (2*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A000108(k)/(n-k)!. - Seiichi Manyama, Aug 15 2023

A365010 E.g.f. satisfies A(x) = 1 + x*exp(-x)*A(x)^3.

Original entry on oeis.org

1, 1, 4, 39, 596, 12365, 324714, 10329655, 386190328, 16597810233, 806356830230, 43700423019011, 2613919719004692, 171053575111641157, 12156558707970920866, 932424974682447304815, 76772968644326739801584, 6754080601542663692950769
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A295239, A302397, A365011.
Cf. A001764, A364983.

Programs

  • Maple
    A365010 := proc(n)
    add( (-k)^(n-k)*A001764(k)/(n-k)!,k=0..n) ;
    %*n! ;
end proc:
seq(A365010(n),n=0..80); # R. J. Mathar, Aug 16 2023
  • PARI
    a(n) = n!*sum(k=0, n, (-k)^(n-k)*binomial(3*k, k)/((2*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} (-k)^(n-k) * A001764(k)/(n-k)!.

A365011 E.g.f. satisfies A(x) = 1 + x*exp(-x)*A(x)^4.

Original entry on oeis.org

1, 1, 6, 87, 1964, 60325, 2349114, 110922091, 6159510552, 393373489257, 28407518470070, 2289019332293551, 203608076603605860, 19816972252710998989, 2094926215725519979698, 239037380421621120397395, 29281119335188021375533104, 3832665229749097186190010193
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A295239, A302397, A365010.
Cf. A002293, A364987.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (-k)^(n-k) * A002293(k)/(n-k)!.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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