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 1-6 of 6 results.

A346894 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^3 / 3!).

Original entry on oeis.org

1, 0, 0, 1, 6, 25, 110, 721, 6286, 57625, 541470, 5558641, 64351166, 819480025, 11140978030, 160711583761, 2472834185646, 40597082635225, 706816137889790, 12974021811748081, 250395124862965726, 5074637684604691225, 107798916619788396750
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 06 2021

Keywords

Links

Crossrefs

Cf. A000670, A330047, A346895, A346920.
Cf. A000392, A327504, A346922, A353774.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[1/(1 - (Exp[x] - 1)^3/3!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] StirlingS2[k, 3] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(1-(exp(x)-1)^3/3!))) \\ Michel Marcus, Aug 06 2021
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (3*k)!*x^(3*k)/(6^k*prod(j=1, 3*k, 1-j*x)))) \\ Seiichi Manyama, May 07 2022
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, binomial(i, j)*stirling(j, 3, 2)*v[i-j+1])); v; \\ Seiichi Manyama, May 07 2022
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 2)/6^k); \\ Seiichi Manyama, May 07 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,3) * a(n-k).
a(n) ~ n! / (3*(1 + 6^(-1/3)) * log(1 + 6^(1/3))^(n+1)). - Vaclav Kotesovec, Aug 08 2021
From Seiichi Manyama, May 07 2022: (Start)
G.f.: Sum_{k>=0} (3*k)! * x^(3*k)/(6^k * Product_{j=1..3*k} (1 - j * x)).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling2(n,3*k)/6^k. (End)

A353664 Expansion of e.g.f. exp((exp(x) - 1)^3).

Original entry on oeis.org

1, 0, 0, 6, 36, 150, 900, 9366, 101556, 1031190, 10995300, 134640726, 1844184276, 26656678230, 400614423300, 6347263038486, 106960986110196, 1905688502565270, 35546025523227300, 691014283378745046, 13999772792477879316, 295570215436360196310
Offset: 0

Views

Author

Seiichi Manyama, May 07 2022

Keywords

Crossrefs

Cf. A000110, A052859, A353665.
Cf. A327504, A353344, A353774.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((exp(x)-1)^3)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (3*k)!*x^(3*k)/(k!*prod(j=1, 3*k, 1-j*x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=6*sum(j=1, i, binomial(i-1, j-1)*stirling(j, 3, 2)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 2)/k!);

Formula

G.f.: Sum_{k>=0} (3*k)! * x^(3*k)/(k! * Product_{j=1..3*k} (1 - j * x)).
a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n-1,k-1) * Stirling2(k,3) * a(n-k).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling2(n,3*k)/k!.

A353775 Expansion of e.g.f. 1/(1 - (exp(x) - 1)^4).

Original entry on oeis.org

1, 0, 0, 0, 24, 240, 1560, 8400, 81144, 1638000, 31058520, 482499600, 6905646264, 114015261360, 2456232531480, 59734751403600, 1427946773067384, 33377481440110320, 818549745973204440, 22338800420915168400, 667566534457962216504, 20735588176755396824880
Offset: 0

Views

Author

Seiichi Manyama, May 07 2022

Keywords

Links

Crossrefs

Cf. A000670, A052841, A353774.
Cf. A346895, A353119, A353665.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1/(1-(Exp[x]-1)^4),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 05 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(exp(x)-1)^4)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (4*k)!*x^(4*k)/prod(j=1, 4*k, 1-j*x)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=24*sum(j=1, i, binomial(i, j)*stirling(j, 4, 2)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\4, (4*k)!*stirling(n, 4*k, 2));

Formula

G.f.: Sum_{k>=0} (4*k)! * x^(4*k)/Product_{j=1..4*k} (1 - j * x).
a(0) = 1; a(n) = 24 * Sum_{k=1..n} binomial(n,k) * Stirling2(k,4) * a(n-k).
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * Stirling2(n,4*k).
a(n) ~ n! / (8 * log(2)^(n+1)). - Vaclav Kotesovec, May 08 2022

A316748 Stirling transform of (3*n)!.

Original entry on oeis.org

1, 6, 726, 365046, 481183926, 1312473466806, 6422019989033526, 51225575261701080246, 621880652519326246083126, 10911229213845806303174823606, 265743324574322126992546955062326, 8697919110119969555113124407898635446, 372566878251517048881238923757823056246326
Offset: 0

Views

Author

Vaclav Kotesovec, Jul 12 2018

Keywords

Links

Crossrefs

Cf. A000670, A064618, A316747, A353774.

Programs

  • Mathematica
    Table[Sum[StirlingS2[n, k]*(3*k)!, {k, 0, n}], {n, 0, 15}]
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (3*k)!*(exp(x)-1)^k/k!))) \\ Seiichi Manyama, May 21 2022

Formula

a(n) ~ (3*n)!.
a(n) ~ sqrt(2*Pi) * 3^(3*n + 1/2) * n^(3*n + 1/2) / exp(3*n).
E.g.f.: Sum_{k>=0} (3*k)! * (exp(x) - 1)^k / k!. - Seiichi Manyama, May 21 2022

A357868 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* Stirling2(n,k*j).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 0, 2, 13, 0, 1, 0, 0, 6, 75, 0, 1, 0, 0, 6, 38, 541, 0, 1, 0, 0, 0, 36, 270, 4683, 0, 1, 0, 0, 0, 24, 150, 2342, 47293, 0, 1, 0, 0, 0, 0, 240, 1260, 23646, 545835, 0, 1, 0, 0, 0, 0, 120, 1560, 16926, 272918, 7087261, 0, 1, 0, 0, 0, 0, 0, 1800, 8400, 197316, 3543630, 102247563, 0
Offset: 0

Views

Author

Seiichi Manyama, Oct 17 2022

Keywords

Examples

			Square array begins:
  1,   1,   1,   1,   1,   1, ...
  0,   1,   0,   0,   0,   0, ...
  0,   3,   2,   0,   0,   0, ...
  0,  13,   6,   6,   0,   0, ...
  0,  75,  38,  36,  24,   0, ...
  0, 541, 270, 150, 240, 120, ...

Crossrefs

Columns k=0-4 give: A000007, A000670, A052841, A353774, A353775.
Cf. A324162, A357293, A357869, A357881.

Programs

  • PARI
    T(n, k) = sum(j=0, n, (k*j)!*stirling(n, k*j, 2));
  • PARI
    T(n, k) = if(k==0, 0^n, n!*polcoef(1/(1-(exp(x+x*O(x^n))-1)^k), n));

Formula

For k > 0, e.g.f. of column k: 1/(1 - (exp(x) - 1)^k).
T(0,k) = 1; T(n,k) = k! * Sum_{j=1..n} binomial(n,j) * Stirling2(j,k) * T(n-j,k).

A373940 Expansion of e.g.f. 1/(1 - (exp(x) - 1)^5).

Original entry on oeis.org

1, 0, 0, 0, 0, 120, 1800, 16800, 126000, 834120, 8731800, 229191600, 6352632000, 143603580120, 2736395461800, 47283190718400, 860150574738000, 20236134851478120, 614854122909391800, 19930647062659477200, 615406024970593164000, 17883373100352330768120
Offset: 0

Views

Author

Seiichi Manyama, Aug 27 2024

Keywords

Crossrefs

Cf. A000670, A052841, A353774, A353775.
Cf. A353200.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(exp(x)-1)^5)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=120*sum(j=1, i, binomial(i, j)*stirling(j, 5, 2)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2));

Formula

G.f.: Sum_{k>=0} (5*k)! * x^(5*k)/Product_{j=1..5*k} (1 - j * x).
a(0) = 1; a(n) = 120 * Sum_{k=1..n} binomial(n,k) * Stirling2(k,5) * a(n-k).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling2(n,5*k).
a(n) ~ n! / (10 * log(2)^(n+1)). - Vaclav Kotesovec, Aug 27 2024
Showing 1-6 of 6 results.

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