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 1-9 of 9 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)

A327505 Number of set partitions of [n] where each subset is again partitioned into four nonempty subsets.

Original entry on oeis.org

1, 0, 0, 0, 1, 10, 65, 350, 1736, 9030, 60355, 561550, 6183221, 69469400, 761767370, 8239194600, 91058524831, 1073790441370, 13900626022985, 196759304278250, 2963381404815566, 46227649788125190, 736940002561065325, 12005645243802471250, 201482801573414254301
Offset: 0

Views

Author

Alois P. Heinz, Sep 14 2019

Keywords

Links

Crossrefs

Column k=4 of A324162.
Cf. A346895.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
      *binomial(n-1, j-1)*Stirling2(j, 4), j=4..n))
    end:
seq(a(n), n=0..25);
  • Mathematica
    a[n_] := a[n] = If[n == 0, 1, Sum[a[n - j] Binomial[n - 1, j - 1] StirlingS2[j, 4], {j, 4, n}]];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 16 2020, after Alois P. Heinz *)
  • PARI
    a(n) = sum(k=0, n\4, (4*k)!*stirling(n, 4*k, 2)/(24^k*k!)); \\ Seiichi Manyama, May 07 2022

Formula

E.g.f.: exp((exp(x)-1)^4/4!).
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * Stirling2(n,4*k)/(24^k * k!). - Seiichi Manyama, May 07 2022

A330047 Expansion of e.g.f. exp(-x) / (1 - sinh(x)).

Original entry on oeis.org

1, 0, 1, 3, 13, 75, 511, 4053, 36793, 375735, 4262971, 53203953, 724379173, 10684377795, 169713810631, 2888340723453, 52433443111153, 1011340189494255, 20654264750645491, 445249365444296553, 10103533212012216733, 240731286454287293115, 6008902898851584479551
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 28 2019

Keywords

Comments

Inverse binomial transform of A006154.

Links

Crossrefs

Cf. A006154, A009227, A052841, A330046, A346894, A346895, A346921.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[-x]/(1 - Sinh[x]), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(exp(x)-1)^2/2))) \\ Seiichi Manyama, May 07 2022
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (2*k)!*x^(2*k)/(2^k*prod(j=1, 2*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, 2, 2)*v[i-j+1])); v; \\ Seiichi Manyama, May 07 2022
  • PARI
    a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 2)/2^k); \\ Seiichi Manyama, May 07 2022

Formula

a(n) = Sum_{k=0..n} (-1)^(n - k) * binomial(n,k) * A006154(k).
a(n) ~ n! / ((2 + sqrt(2)) * (log(1 + sqrt(2)))^(n+1)). - Vaclav Kotesovec, Dec 03 2019
From Seiichi Manyama, May 07 2022: (Start)
E.g.f.: 1/(1 - (exp(x) - 1)^2 / 2).
G.f.: Sum_{k>=0} (2*k)! * x^(2*k)/(2^k * Product_{j=1..2*k} (1 - j * x)).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,2) * a(n-k).
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * Stirling2(n,2*k)/2^k. (End)
a(0) = 1; a(n) = (-1)^n + Sum_{k=1..ceiling(n/2)} binomial(n,2*k-1) * a(n-2*k+1). - Prabha Sivaramannair, Oct 06 2023

A346923 Expansion of e.g.f. 1 / (1 - log(1 - x)^4 / 4!).

Original entry on oeis.org

1, 0, 0, 0, 1, 10, 85, 735, 6839, 69804, 784580, 9680000, 130312336, 1901581968, 29895585356, 503657235900, 9051009737834, 172807817059664, 3493189152511608, 74530548004474584, 1673793045085649146, 39467836062718058100, 974939402596817961050, 25177327470510057799550
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 07 2021

Keywords

Links

Crossrefs

Cf. A007840, A346921, A346922, A346924.
Cf. A000454, A346895, A347003, A353119.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[1/(1 - Log[1 - x]^4/4!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Abs[StirlingS1[k, 4]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(1-log(1-x)^4/4!))) \\ Michel Marcus, Aug 07 2021
  • PARI
    a(n) = sum(k=0, n\4, (4*k)!*abs(stirling(n, 4*k, 1))/24^k); \\ Seiichi Manyama, May 06 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * |Stirling1(k,4)| * a(n-k).
a(n) ~ n! * 2^(-5/4) * 3^(1/4) / (exp(2^(3/4)*3^(1/4)) * (1 - exp(-2^(3/4)*3^(1/4)))^(n+1)). - Vaclav Kotesovec, Aug 08 2021
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * |Stirling1(n,4*k)|/24^k. - Seiichi Manyama, May 06 2022

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

A346920 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^5 / 5!).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 15, 140, 1050, 6951, 42777, 260590, 1809060, 17418401, 229768539, 3402511476, 50013258750, 706670789371, 9659104177101, 130958047050698, 1834295186003784, 27849428308615221, 472297857494304303, 8856291348143365456, 176841068643273207426
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 07 2021

Keywords

Links

Crossrefs

Cf. A000670, A330047, A346894, A346895.
Cf. A000481, A327506, A346924.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[1/(1 - (Exp[x] - 1)^5/5!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] StirlingS2[k, 5] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(1-(exp(x)-1)^5/5!))) \\ Michel Marcus, Aug 07 2021
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (5*k)!*x^(5*k)/(120^k*prod(j=1, 5*k, 1-j*x)))) \\ Seiichi Manyama, May 09 2022
  • PARI
    a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2)/120^k); \\ Seiichi Manyama, May 09 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,5) * a(n-k).
a(n) ~ n! / (5*(1 + 120^(-1/5)) * log(1 + 120^(1/5))^(n+1)). - Vaclav Kotesovec, Aug 08 2021
From Seiichi Manyama, May 09 2022: (Start)
G.f.: Sum_{k>=0} (5*k)! * x^(5*k)/(120^k * Product_{j=1..5*k} (1 - j * x)).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling2(n,5*k)/120^k. (End)

A354393 Expansion of e.g.f. 1/(1 + (exp(x) - 1)^4 / 24).

Original entry on oeis.org

1, 0, 0, 0, -1, -10, -65, -350, -1631, -5250, 18395, 685850, 10485739, 127737610, 1336804105, 11432407350, 54280609109, -712071643930, -29671691715185, -660215774400350, -11770593620859521, -176475952496559870, -2055362595355830475, -9749893741512339250
Offset: 0

Views

Author

Seiichi Manyama, May 25 2022

Keywords

Crossrefs

Cf. A354391, A354392, A354394.
Cf. A346895, A354390, A354397.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+(exp(x)-1)^4/24)))
  • 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, 4, 2)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\4, (4*k)!*stirling(n, 4*k, 2)/(-24)^k);

Formula

a(0) = 1; a(n) = -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)/(-24)^k.

A353885 Expansion of e.g.f. 1/(1 - (x * (exp(x) - 1))^4 / 576).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 70, 1260, 13650, 115500, 841995, 5555550, 34139105, 198948750, 1175994820, 10315705400, 192609389700, 4563951046200, 98992258506345, 1898260633492650, 32787422848455275, 520556451785466250, 7722233521138092726, 108688302800107222500
Offset: 0

Views

Author

Seiichi Manyama, May 09 2022

Keywords

Crossrefs

Cf. A052848, A353883, A353884.
Cf. A346895, A353882.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(x*(exp(x)-1))^4/576)))
  • PARI
    a(n) = n!*sum(k=0, n\8, (4*k)!*stirling(n-4*k, 4*k, 2)/(576^k*(n-4*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/8)} (4*k)! * Stirling2(n-4*k,4*k)/(576^k * (n-4*k)!).

A346954 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^4 / 4! ).

Original entry on oeis.org

1, 10, 65, 350, 1736, 9030, 60355, 561550, 6188996, 69919850, 781211795, 8854058850, 106994019406, 1433756147470, 21287253921635, 339206526695750, 5630710652048216, 96341917117951890, 1708973354556320875, 31787279786739738250, 623964823224788294426
Offset: 4

Views

Author

Ilya Gutkovskiy, Aug 08 2021

Keywords

Crossrefs

Cf. A000453, A000629, A003704, A327505, A346390, A346895, A346955.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[-Log[1 - (Exp[x] - 1)^4/4!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 4] &
a[n_] := a[n] = StirlingS2[n, 4] + (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 4] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 4, 24}]

Formula

a(n) = Stirling2(n,4) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,4) * k * a(k).
a(n) ~ (n-1)! / (log(2^(3/4)*3^(1/4) + 1))^n. - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/4)} (4*k)! * Stirling2(n,4*k)/(k * 24^k). - Seiichi Manyama, Jan 23 2025
Showing 1-9 of 9 results.

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

Content is available under The OEIS End-User License Agreement.