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.

A351503 Expansion of e.g.f. 1/(1 + x^2 * log(1 - x)).

Original entry on oeis.org

1, 0, 0, 6, 12, 40, 900, 6048, 43680, 717120, 8658720, 102231360, 1735525440, 28819964160, 473955850368, 9235543363200, 189202617676800, 3940225003653120, 89804740509434880, 2169337606086389760, 54085753764912844800, 1429100881569205125120
Offset: 0

Views

Author

Seiichi Manyama, May 04 2022

Keywords

Links

Crossrefs

Cf. A052830, A351504.
Cf. A351505, A353228.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1/(1+x^2 Log[1-x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 18 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x^2*log(1-x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!*sum(j=3, i, 1/(j-2)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\3, k!*abs(stirling(n-2*k, k, 1))/(n-2*k)!);

Formula

a(0) = 1; a(n) = n! * Sum_{k=3..n} 1/(k-2) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/3)} k! * |Stirling1(n-2*k,k)|/(n-2*k)!.

A351506 Expansion of e.g.f. 1/(1 + x^3/6 * log(1 - x)).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 40, 210, 2464, 20160, 178800, 1755600, 21215040, 268107840, 3596916960, 51452200800, 800489733120, 13262804755200, 232536822336000, 4300843392518400, 84023034413644800, 1727339274045504000, 37248117171719731200, 840387048760633651200
Offset: 0

Views

Author

Seiichi Manyama, May 04 2022

Keywords

Links

Crossrefs

Cf. A052830, A351505.
Cf. A351493, A351504,

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x^3/6*log(1-x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!/6*sum(j=4, i, 1/(j-3)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\4, k!*abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!));

Formula

a(0) = 1; a(n) = n!/6 * Sum_{k=4..n} 1/(k-3) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/4)} k! * |Stirling1(n-3*k,k)|/(6^k * (n-3*k)!).

A353229 Expansion of e.g.f. (1 - x)^(-x^3).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 240, 1260, 28224, 241920, 2181600, 21621600, 315342720, 4358914560, 61607407680, 912518006400, 15142006978560, 265601118182400, 4877947688140800, 93691850626483200, 1901787789077452800, 40548028309147699200, 904101131200045363200
Offset: 0

Views

Author

Seiichi Manyama, May 01 2022

Keywords

Links

Crossrefs

Cf. A066166, A351493, A351504, A353228.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[(1-x)^-x^3,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 20 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(-x^3)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x^3*log(1-x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=4, i, j/(j-3)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\4, abs(stirling(n-3*k, k, 1))/(n-3*k)!);

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=4..n} k/(k-3) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/4)} |Stirling1(n-3*k,k)|/(n-3*k)!.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / exp(n). - Vaclav Kotesovec, May 04 2022

A358014 Expansion of e.g.f. 1/(1 - x^3 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 120, 210, 40656, 363384, 2117520, 9980190, 520250280, 9496208436, 109522054824, 982593614730, 28426015541280, 762523155318000, 14192088961120416, 204618562767970614, 4906638448867994040, 154037798077765359660, 4000484484370905087480
Offset: 0

Views

Author

Seiichi Manyama, Oct 24 2022

Keywords

Crossrefs

Cf. A052848, A358013.
Cf. A292891, A351504, A353999.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1/(1-x^3 (Exp[x]-1)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 26 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^3*(exp(x)-1))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!*sum(j=4, i, 1/(j-3)!*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\4, k!*stirling(n-3*k, k, 2)/(n-3*k)!);

Formula

a(0) = 1; a(n) = n! * Sum_{k=4..n} 1/(k-3)! * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/4)} k! * Stirling2(n-3*k,k)/(n-3*k)!.

A375701 Expansion of e.g.f. 1 / sqrt(1 + x^3 * log(1 - x)).

Original entry on oeis.org

1, 0, 0, 0, 12, 30, 120, 630, 19152, 166320, 1506600, 14968800, 313014240, 4864860000, 72829607760, 1116874558800, 23605893400320, 495461472105600, 10289649464640000, 215706738207542400, 5222625647551920000, 133507746422859513600, 3481696859911699968000
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2024

Keywords

Crossrefs

Cf. A351504, A375699, A375700.
Cf. A375718.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1+x^3*log(1-x))))
  • PARI
    a(n) = n!*sum(k=0, n\4, prod(j=0, k-1, 6*j+3)*abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (Product_{j=0..k-1} (6*j+3)) * |Stirling1(n-3*k,k)|/(6^k*(n-3k)!).

A370995 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + x^3*log(1-x)) ).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 240, 1260, 209664, 2056320, 20476800, 221205600, 19370292480, 406935809280, 7376151444480, 131868581644800, 8376837844193280, 282378273124147200, 7891890567682867200, 207283550601631795200, 11520967360247698636800
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A052802, A370993, A370994.
Cf. A351504.

Programs

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

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} (n+k)! * |Stirling1(n-3*k,k)|/(n-3*k)!.

A375699 Expansion of e.g.f. 1 / (1 + x^3 * log(1 - x))^(1/6).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 40, 210, 5264, 45360, 409800, 4065600, 77948640, 1183422240, 17527233360, 267109642800, 5422495921920, 110998923235200, 2270809072896000, 47142009514454400, 1116394268619772800, 27963045712157472000, 718066383283082803200
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2024

Keywords

Crossrefs

Cf. A351504, A375700, A375701.
Cf. A008542, A351506.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x^3*log(1-x))^(1/6)))
  • PARI
    a(n) = n!*sum(k=0, n\4, prod(j=0, k-1, 6*j+1)*abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (Product_{j=0..k-1} (6*j+1)) * |Stirling1(n-3*k,k)|/(6^k*(n-3k)!).

A375700 Expansion of e.g.f. 1 / (1 + x^3 * log(1 - x))^(1/3).

Original entry on oeis.org

1, 0, 0, 0, 8, 20, 80, 420, 11648, 100800, 912000, 9055200, 181547520, 2790627840, 41568334080, 635617382400, 13172198645760, 273158953267200, 5632405756723200, 117530452124467200, 2815021136030515200, 71252240659839590400, 1844362570865444044800
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2024

Keywords

Crossrefs

Cf. A351504, A375699, A375701.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x^3*log(1-x))^(1/3)))
  • PARI
    a(n) = n!*sum(k=0, n\4, prod(j=0, k-1, 6*j+2)*abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (Product_{j=0..k-1} (6*j+2)) * |Stirling1(n-3*k,k)|/(6^k*(n-3k)!).

A355665 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + x^k * log(1 - x)).

Original entry on oeis.org

1, 1, 1, 1, 0, 3, 1, 0, 2, 14, 1, 0, 0, 3, 88, 1, 0, 0, 6, 32, 694, 1, 0, 0, 0, 12, 150, 6578, 1, 0, 0, 0, 24, 40, 1524, 72792, 1, 0, 0, 0, 0, 60, 900, 12600, 920904, 1, 0, 0, 0, 0, 120, 240, 6048, 147328, 13109088, 1, 0, 0, 0, 0, 0, 360, 1260, 43680, 1705536, 207360912
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2022

Keywords

Examples

			Square array begins:
     1,    1,   1,   1,   1,   1, 1, ...
     1,    0,   0,   0,   0,   0, 0, ...
     3,    2,   0,   0,   0,   0, 0, ...
    14,    3,   6,   0,   0,   0, 0, ...
    88,   32,  12,  24,   0,   0, 0, ...
   694,  150,  40,  60, 120,   0, 0, ...
  6578, 1524, 900, 240, 360, 720, 0, ...

Crossrefs

Columns k=0..3 give A007840, A052830, A351503, A351504.
Cf. A355609, A355652.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n\(k+1), j!*abs(stirling(n-k*j, j, 1))/(n-k*j)!);

Formula

T(0,k) = 1 and T(n,k) = n! * Sum_{j=k+1..n} 1/(j-k) * T(n-j,k)/(n-j)! for n > 0.
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} j! * |Stirling1(n-k*j,j)|/(n-k*j)!.
Showing 1-9 of 9 results.

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

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