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 1-7 of 7 results.

A346924 Expansion of e.g.f. 1 / (1 + log(1 - x)^5 / 5!).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 15, 175, 1960, 22449, 269577, 3430790, 46480830, 671260876, 10329270952, 169125055736, 2940784282800, 54182845939104, 1055291277366108, 21674715826211532, 468366193441002564, 10624074081842024496, 252432685158931968768, 6270222495850552958004
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 07 2021

Keywords

Links

Crossrefs

Cf. A007840, A346921, A346922, A346923.
Cf. A000482, A347004, A353200.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[1/(1 + Log[1 - x]^5/5!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Abs[StirlingS1[k, 5]] 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)^5/5!))) \\ Michel Marcus, Aug 07 2021
  • PARI
    a(n) = sum(k=0, n\5, (5*k)!*abs(stirling(n, 5*k, 1))/120^k); \\ Seiichi Manyama, May 06 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * |Stirling1(k,5)| * a(n-k).
a(n) ~ n! * 2^(3/5) * 3^(1/5) * exp(2^(3/5)*15^(1/5)*n) / (5^(4/5) * (exp(2^(3/5)*15^(1/5)) - 1)^(n+1)). - Vaclav Kotesovec, Aug 08 2021
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * |Stirling1(n,5*k)|/120^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

A353119 Expansion of e.g.f. 1/(1 - log(1 - x)^4).

Original entry on oeis.org

1, 0, 0, 0, 24, 240, 2040, 17640, 202776, 3066336, 52446720, 933636000, 17416490784, 350580364992, 7719355635264, 184232862777600, 4691944607751936, 126358891201529856, 3591751011211717632, 107772466927523060736, 3408777017097439186944
Offset: 0

Views

Author

Seiichi Manyama, May 06 2022

Keywords

Links

Crossrefs

Cf. A052811, A353118, A353200.
Cf. A346923.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(1-Log[1-x]^4),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 13 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1-x)^4)))
  • 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)*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)));

Formula

a(0) = 1; a(n) = 24 * Sum_{k=1..n} binomial(n,k) * |Stirling1(k,4)| * a(n-k).
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * |Stirling1(n,4*k)|.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (4 * (exp(1) - 1)^(n+1)). - Vaclav Kotesovec, May 07 2022

A353404 Expansion of e.g.f. exp(-log(1 - x)^5).

Original entry on oeis.org

1, 0, 0, 0, 0, 120, 1800, 21000, 235200, 2693880, 34133400, 509823600, 9012207600, 180053908320, 3870208261920, 87083930169600, 2034907999488000, 49491370609706880, 1259740748821328640, 33710658096392887680, 949893399326820528000
Offset: 0

Views

Author

Seiichi Manyama, May 06 2022

Keywords

Links

Crossrefs

Column k=5 of A357882.
Cf. A000142, A353344, A353358.
Cf. A347004, A353200.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[-Log[1-x]^5],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 13 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-log(1-x)^5)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(-log(1-x)^4)))
  • 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-1, j-1)*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))/k!);

Formula

E.g.f.: (1 - x)^(-(log(1 - x))^4).
a(0) = 1; a(n) = 120 * Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,5)| * a(n-k).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * |Stirling1(n,5*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).

A354230 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, -28690200, 210447600, 1465952400, -123513355680, 4155643171680, -114924516470400, 2886135295680000, -66750668391381120, 1375830884058456960, -22036006671394705920, 70186623981895296000, 16180846322732941893120
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Crossrefs

Cf. A006252, A354229.
Cf. A353200, A354135, A354232.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1/(1-Log[1+x]^5),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 20 2024 *)
  • 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)*stirling(j, 5, 1)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\5, (5*k)!*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).

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 1-7 of 7 results.

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