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 1-2 of 2 results.

A355163 a(n) = exp(-1) * Sum_{k>=0} (4*k + 3)^n / k!.

Original entry on oeis.org

1, 7, 65, 743, 9921, 150151, 2526593, 46615783, 933072513, 20093861895, 462440842177, 11310514854375, 292627518129985, 7976748158144647, 228308400790500097, 6840702405678586343, 214000748166439723265, 6973447420429351808007, 236204029044752265931585, 8300724166287243795922151
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 22 2022

Keywords

Crossrefs

Cf. A000110, A005494, A126390, A284859, A284864, A285064, A355162.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[Exp[Exp[4 x] + 3 x - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 3 a[n - 1] + Sum[Binomial[n - 1, k - 1] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
Table[Sum[Binomial[n, k] 3^(n - k) 4^k BellB[k], {k, 0, n}], {n, 0, 19}]

Formula

E.g.f.: exp(exp(4*x) + 3 x - 1).
a(0) = 1; a(n) = 3 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 4^k * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * 3^(n-k) * 4^k * Bell(k).
a(n) ~ Bell(n) * (4 + 3*LambertW(n)/n)^n. - Vaclav Kotesovec, Jun 22 2022
a(n) ~ 4^n * n^(n + 3/4) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n + 3/4)). - Vaclav Kotesovec, Jun 27 2022

A367939 Expansion of e.g.f. exp(exp(4*x) - 1 - 2*x).

Original entry on oeis.org

1, 2, 20, 168, 1936, 25376, 378688, 6284928, 114471168, 2263605760, 48192279552, 1097180784640, 26562251100160, 680591327567872, 18381995707154432, 521521320660205568, 15495495061984051200, 480873815489757970432, 15549555768325162926080, 522810678067316117733376
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 05 2023

Keywords

Crossrefs

Cf. A000110, A124311, A126617, A355162, A367786, A367938, A367940.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[Exp[Exp[4 x] - 1 - 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -2 a[n - 1] + Sum[Binomial[n - 1, k - 1] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
Table[Sum[Binomial[n, k] (-2)^(n + k) BellB[k], {k, 0, n}], {n, 0, 19}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(exp(4*x) - 1 - 2*x))) \\ Michel Marcus, Dec 07 2023

Formula

G.f. A(x) satisfies: A(x) = 1 - 2 * x * ( A(x) - 2 * A(x/(1 - 4*x)) / (1 - 4*x) ).
a(n) = exp(-1) * Sum_{k>=0} (4*k-2)^n / k!.
a(0) = 1; a(n) = -2 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 4^k * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-2)^(n+k) * Bell(k).
a(n) = 2^n * |A124311(n)|.
Showing 1-2 of 2 results.

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

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