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.

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

A353118 Expansion of e.g.f. 1/(1 + log(1 - x)^3).

Original entry on oeis.org

1, 0, 0, 6, 36, 210, 2070, 24864, 310632, 4337544, 68922360, 1205002656, 22844264256, 469287123552, 10397824478496, 246800350393344, 6246190572981120, 167972669001740160, 4783274802508890240, 143775432034543203840, 4548946867429143444480
Offset: 0

Views

Author

Seiichi Manyama, May 06 2022

Keywords

Links

Crossrefs

Cf. A052811, A353119, A353200.
Cf. A346922.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(1+Log[1-x]^3),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 04 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x)^3)))
  • 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, j)*abs(stirling(j, 3, 1))*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*abs(stirling(n, 3*k, 1)));

Formula

a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n,k) * |Stirling1(k,3)| * a(n-k).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * |Stirling1(n,3*k)|.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (3 * (exp(1) - 1)^(n+1)). - Vaclav Kotesovec, May 07 2022

A353200 Expansion of e.g.f. 1/(1 + log(1 - x)^5).

Original entry on oeis.org

1, 0, 0, 0, 0, 120, 1800, 21000, 235200, 2693880, 35947800, 609615600, 12504927600, 281242996320, 6545492073120, 155873050569600, 3849612346944000, 100588974863402880, 2818516832681523840, 84728757269204858880, 2706516690047188416000
Offset: 0

Views

Author

Seiichi Manyama, May 06 2022

Keywords

Links

Crossrefs

Cf. A052811, A353118, A353119.
Cf. A346924.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x)^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)*abs(stirling(j, 5, 1))*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\5, (5*k)!*abs(stirling(n, 5*k, 1)));

Formula

a(0) = 1; a(n) = 120 * Sum_{k=1..n} binomial(n,k) * |Stirling1(k,5)| * a(n-k).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * |Stirling1(n,5*k)|.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (5 * (exp(1) - 1)^(n+1)). - Vaclav Kotesovec, May 07 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

A353358 Expansion of e.g.f. exp(log(1 - x)^4).

Original entry on oeis.org

1, 0, 0, 0, 24, 240, 2040, 17640, 182616, 2340576, 34907520, 567732000, 9811675104, 179804319552, 3507724531584, 72964001073600, 1614757714491456, 37860036000293376, 936291898320463872, 24333527620574701056, 662723505438520771584, 18871765275000834201600
Offset: 0

Views

Author

Seiichi Manyama, May 06 2022

Keywords

Links

Crossrefs

Column k=4 of A357882.
Cf. A000142, A353344, A353404.
Cf. A347003, A353119, A353665.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(log(1-x)^4)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(log(1-x)^3)))
  • 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-1, j-1)*abs(stirling(j, 4, 1))*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\4, (4*k)!*abs(stirling(n, 4*k, 1))/k!);

Formula

E.g.f.: (1 - x)^((log(1 - x))^3).
a(0) = 1; a(n) = 24 * Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,4)| * a(n-k).
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * |Stirling1(n,4*k)|/k!.

A357881 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* |Stirling1(n,k*j)|.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 0, 2, 14, 0, 1, 0, 0, 6, 88, 0, 1, 0, 0, 6, 46, 694, 0, 1, 0, 0, 0, 36, 340, 6578, 0, 1, 0, 0, 0, 24, 210, 3308, 72792, 0, 1, 0, 0, 0, 0, 240, 2070, 36288, 920904, 0, 1, 0, 0, 0, 0, 120, 2040, 24864, 460752, 13109088, 0, 1, 0, 0, 0, 0, 0, 1800, 17640, 310632, 6551424, 207360912, 0
Offset: 0

Views

Author

Seiichi Manyama, Oct 18 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,  14,   6,   6,   0,   0, ...
  0,  88,  46,  36,  24,   0, ...
  0, 694, 340, 210, 240, 120, ...

Crossrefs

Columns k=0-5 give: A000007, A007840, A052811, A353118, A353119, A353200.
Cf. A357119, A357868, A357882.

Programs

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

Formula

For k > 0, e.g.f. of column k: 1/(1 - (-log(1-x))^k).
T(0,k) = 1; T(n,k) = k! * Sum_{j=1..n} binomial(n,j) * |Stirling1(j,k)| * T(n-j,k).
Showing 1-6 of 6 results.

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