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.

A351274 a(0) = 1; thereafter a(n) = Sum_{k=1..n} (2*k)^k * Stirling1(n,k).

Original entry on oeis.org

1, 2, 14, 172, 2964, 65848, 1789688, 57521280, 2133964352, 89744964288, 4219022123328, 219246630903936, 12479659844383104, 772174659456713472, 51603153976362554112, 3704166182571098222592, 284239227254465994240000, 23218955083323248158556160
Offset: 0

Views

Author

Seiichi Manyama, Feb 05 2022

Keywords

Links

Crossrefs

Cf. A305819, A351182, A351275, A351276.

Programs

  • Mathematica
    Join[{1},Table[Sum[(2k)^k StirlingS1[n,k],{k,n}],{n,20}]] (* Harvey P. Dale, Dec 31 2023 *)
  • PARI
    a(n) = sum(k=0, n, (2*k)^k*stirling(n, k, 1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-2*log(1+x)))))

Formula

E.g.f.: 1/(1 + LambertW( -2 * log(1+x) )), where LambertW() is the Lambert W-function.
a(n) ~ n^n / (sqrt(2) * (exp(exp(-1)/2) - 1)^(n+1/2) * exp(n - exp(-1)/4 + 1/2)). - Vaclav Kotesovec, Feb 06 2022

Extensions

Made a(0) = 1 explicit and changed range of k in definition to start at 1 at the suggestion of Harvey P. Dale. - N. J. A. Sloane, Dec 31 2023

A351277 a(n) = Sum_{k=0..n} (-2*k)^k * Stirling2(n,k).

Original entry on oeis.org

1, -2, 14, -170, 2910, -64202, 1733278, -55338250, 2039421598, -85204516298, 3979272245662, -205432301027978, 11616783053131934, -714082744228546890, 47409028234931260318, -3380871137079666543114, 257736986308003127354014
Offset: 0

Views

Author

Seiichi Manyama, Feb 05 2022

Keywords

Links

Crossrefs

Cf. A351218, A351275, A351276.

Programs

  • PARI
    a(n) = sum(k=0, n, (-2*k)^k*stirling(n, k, 2));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(2*(exp(x)-1)))))

Formula

E.g.f.: 1/(1 + LambertW( 2 * (exp(x) - 1) )), where LambertW() is the Lambert W-function.
a(n) ~ (-1)^n * n^n / (sqrt(2*exp(1) - 1) * exp(n) * (1 - log(exp(1) - 1/2))^(n + 1/2)). - Vaclav Kotesovec, Feb 06 2022

A355781 E.g.f. satisfies log(A(x)) = 2 * (exp(x) - 1) * A(x).

Original entry on oeis.org

1, 2, 14, 166, 2854, 64854, 1839622, 62688406, 2497159302, 113932356630, 5860555367814, 335639363668118, 21184456464757894, 1461163816568091926, 109351697864286862214, 8825909581376322510230, 764231343305480319046278, 70670539764733828998689302
Offset: 0

Views

Author

Seiichi Manyama, Jul 16 2022

Keywords

Links

Crossrefs

Cf. A052880, A351276, A355788.

Programs

  • Maple
    b:= proc(n, m) option remember; `if`(n=0,
     2^m*(m+1)^(m-1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..21);  # Alois P. Heinz, Jul 29 2022
  • Mathematica
     b[n_, m_] := b[n, m] = If[n == 0, 2^m*(m + 1)^(m - 1), m*b[n - 1, m] + b[n - 1, m + 1]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 16 2022, after Alois P. Heinz *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(2*(1-exp(x))))))
  • PARI
    a(n) = sum(k=0, n, 2^k*(k+1)^(k-1)*stirling(n, k, 2));

Formula

E.g.f.: exp( -LambertW(2 * (1 - exp(x))) ).
a(n) = Sum_{k=0..n} 2^k * (k+1)^(k-1) * Stirling2(n,k).
From Vaclav Kotesovec, Jul 18 2022: (Start)
E.g.f.: LambertW(2 * (1 - exp(x))) / (2 * (1 - exp(x))).
a(n) ~ sqrt(2*exp(1) + 1) * sqrt(log(1 + exp(-1)/2)) * n^(n-1) / (exp(n-1) * (log(exp(1) + 1/2) - 1)^n). (End)
Showing 1-3 of 3 results.

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

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