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.

A351135 a(n) = Sum_{k=0..n} k! * k^(k*n) * Stirling1(n,k).

Original entry on oeis.org

1, 1, 31, 117716, 103060088854, 35762522985456876854, 7426384178533125493811949517898, 1294894823429942179301223205449027573956692920, 253092741940931724343266089700550691376738432767085871485096840
Offset: 0

Views

Author

Seiichi Manyama, Feb 02 2022

Keywords

Links

Crossrefs

Cf. A006252, A320083, A351133, A351134.
Cf. A350721, A351117, A351138.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[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, 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

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

A351137 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(3*n) * Stirling1(n,k).

Original entry on oeis.org

1, 1, 129, 121172, 421875178, 3922823960054, 80130334773241142, 3156849112458066440568, 218554371053209725986724984, 24795129220015277612148345850896, 4365539219231132131300647267518575008, 1141930521329052244894253748456776246166288
Offset: 0

Views

Author

Seiichi Manyama, Feb 02 2022

Keywords

Comments

In general, for m >= 0, Sum_{k=0..n} (-1)^(n-k) * k! * k^(m*n) * Stirling1(n,k) ~ c * r^(m*n) * (1 + r*exp(m/r))^n * n^((m+1)*n + 1/2) / exp((m+1)*n), where r is the real root of the equation LambertW(-1, -r*exp(-r)) = -r - exp(-m/r) and c is a constant (depending only on m). - Vaclav Kotesovec, Feb 04 2022

Links

Crossrefs

Cf. A007840 (m=0), A320096 (m=1), A351136 (m=2).
Cf. A203798, A351134, A351138.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[(-1)^(n - k) * 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, (-1)^(n-k)*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 * r^(3*n) * (1 + r*exp(3/r))^n * n^(4*n + 1/2) / exp(4*n), where r = 0.97698437755148201976772582981871258235824532360125531194... is the real root of the equation LambertW(-1, -r*exp(-r)) = -r - exp(-3/r) and c = 2.3655154360078103511101518906595610482889989819... - Vaclav Kotesovec, Feb 04 2022
Showing 1-3 of 3 results.

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