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.

A351136 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(2*n) * Stirling1(n,k).

Original entry on oeis.org

1, 1, 33, 4760, 1814698, 1436035954, 2041681617638, 4736066140912728, 16729538152432476024, 85437808930634601070944, 605822464949212598847700512, 5774077466357788471179323050704, 72030066703292325305595937373723040
Offset: 0

Views

Author

Seiichi Manyama, Feb 02 2022

Keywords

Links

Crossrefs

Cf. A007840, A320096, A351137.
Cf. A187755, A351133, A351138.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[(-1)^(n - k) * k! * k^(2*n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 13, 0] (* Amiram Eldar, Feb 02 2022 *)
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*k!*k^(2*n)*stirling(n, k, 1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-log(1-k^2*x))^k)))

Formula

E.g.f.: Sum_{k>=0} (-log(1 - k^2*x))^k.
a(n) ~ c * r^(2*n) * (1 + r*exp(2/r))^n * n^(3*n + 1/2) / exp(3*n), where r = 0.9414380538633895499299457441124149470954491698433... is the real root of the equation LambertW(-1, -r*exp(-r)) = -r - exp(-2/r) and c = 2.22047212763474863127102273073825610210704559048894... - Vaclav Kotesovec, Feb 03 2022

A351134 a(n) = Sum_{k=0..n} k! * k^(3*n) * Stirling1(n,k).

Original entry on oeis.org

1, 1, 127, 115028, 383611414, 3407421330934, 66396378581670602, 2493320561997330821496, 164454446238949941359354760, 17769323863754938530919641304080, 2978930835291629440372517431365668448, 741834782450714229554166000654848368247568
Offset: 0

Views

Author

Seiichi Manyama, Feb 02 2022

Keywords

Links

Crossrefs

Cf. A006252, A320083, A351133.
Cf. A242229, A351135, A351137.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[k! * k^(3*n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 12, 0] (* Amiram Eldar, Feb 02 2022 *)
  • PARI
    a(n) = sum(k=0, n, k!*k^(3*n)*stirling(n, k, 1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, log(1+k^3*x)^k)))

Formula

E.g.f.: Sum_{k>=0} log(1 + k^3*x)^k.
a(n) ~ c * d^n * n^(4*n + 1/2), where d = 0.358437102792682941192966771107499325675345706113923587904567864366079667... and c = 2.68150179193269103258189978938660205530269361522513... - Vaclav Kotesovec, Feb 04 2022

A351138 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(k*n) * Stirling1(n,k).

Original entry on oeis.org

1, 1, 33, 118484, 103098352618, 35763050751038414134, 7426387531294394110580641088438, 1294894837982331434068068403253026516109577144, 253092742000650212462862632240661689524832716838851180353875064
Offset: 0

Views

Author

Seiichi Manyama, Feb 02 2022

Keywords

Links

Crossrefs

Cf. A007840, A320096, A351136, A351137.
Cf. A249584, A351135.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[(-1)^(n - k) * k! * k^(k*n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 9, 0] (* Amiram Eldar, Feb 02 2022 *)
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*k!*k^(k*n)*stirling(n, k, 1));
  • PARI
    my(N=10, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-log(1-k^k*x))^k)))

Formula

E.g.f.: Sum_{k>=0} (-log(1 - k^k*x))^k.
a(n) ~ n! * n^(n^2). - Vaclav Kotesovec, Feb 03 2022
Showing 1-3 of 3 results.

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