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.

A353228 Expansion of e.g.f. (1 - x)^(-x^2).

Original entry on oeis.org

1, 0, 0, 6, 12, 40, 540, 3528, 25200, 263520, 2741760, 30048480, 372794400, 4971957120, 70612686144, 1076056027200, 17469796780800, 300562292459520, 5468568356666880, 104917700221125120, 2116572758902425600, 44794683422986936320, 992435268252158438400
Offset: 0

Views

Author

Seiichi Manyama, May 01 2022

Keywords

Links

Crossrefs

Cf. A066166, A351503, A351492, A353223, A353229.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1-x)^(-x^2), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, May 12 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(-x^2)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-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-1)!*sum(j=3, i, j/(j-2)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\3, abs(stirling(n-2*k, k, 1))/(n-2*k)!);

Formula

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

A353225 Expansion of e.g.f. (1 - x^4)^(-1/x^3).

Original entry on oeis.org

1, 1, 1, 1, 1, 61, 361, 1261, 3361, 128521, 1678321, 11670121, 56596321, 1773048421, 37020623641, 410615985781, 3056256665281, 88439609228881, 2516514283997281, 39513591769228561, 409546654143301441, 11679302565962651341, 413008783534735181641
Offset: 0

Views

Author

Seiichi Manyama, May 01 2022

Keywords

Crossrefs

Cf. A121452, A353223, A353224.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[(1-x^4)^(-1/x^3),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 17 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^4)^(-1/x^3)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-log(1-x^4)/x^3)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, (i+3)\4, (4*j-3)/j*v[i-4*j+4]/(i-4*j+3)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\4, abs(stirling(n-3*k, n-4*k, 1))/(n-3*k)!);

Formula

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

A353227 Expansion of e.g.f. (1 - x^3)^(-x).

Original entry on oeis.org

1, 0, 0, 0, 24, 0, 0, 2520, 20160, 0, 1209600, 19958400, 79833600, 1556755200, 39956716800, 326918592000, 5056340889600, 148203095040000, 1867358997504000, 30411275102208000, 946128558735360000, 15965919428659200000, 293266062902292480000
Offset: 0

Views

Author

Seiichi Manyama, May 01 2022

Keywords

Crossrefs

Cf. A066166, A351156, A353223, A353226.

Programs

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

Formula

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

A357964 Expansion of e.g.f. exp( (exp(x^3) - 1)/x^2 ).

Original entry on oeis.org

1, 1, 1, 1, 13, 61, 181, 1261, 12601, 77113, 481321, 6102361, 63041221, 492260341, 6041807773, 87670198981, 945716793841, 11365316711281, 193962371184721, 2824572189001393, 36983289122143741, 658584258052917421, 12073641790111934341, 185876257572349699741
Offset: 0

Views

Author

Seiichi Manyama, Oct 22 2022

Keywords

Crossrefs

Cf. A357962, A357965.
Cf. A353223.

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} Stirling2(n-2*k,n-3*k)/(n-2*k)!.

A375799 Expansion of e.g.f. 1/(1 + (log(1 - x^3))/x^2).

Original entry on oeis.org

1, 1, 2, 6, 36, 240, 1800, 16800, 178080, 2086560, 27518400, 399168000, 6286896000, 107623676160, 1984274772480, 39143052748800, 824445099878400, 18450791322163200, 437015358530150400, 10929450232744243200, 287728555881102336000, 7952251084537503744000
Offset: 0

Views

Author

Seiichi Manyama, Aug 29 2024

Keywords

Crossrefs

Cf. A007840, A375798.
Cf. A353223.

Programs

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

Formula

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

A353222 Expansion of e.g.f. (1 - x^3)^(-1/x).

Original entry on oeis.org

1, 0, 2, 0, 12, 60, 120, 2520, 15120, 90720, 1693440, 13305600, 140374080, 2724321600, 27744837120, 414096883200, 8689288608000, 111399326438400, 2114134793971200, 48501156601497600, 759659036405068800, 17279306372135808000, 434100706059205785600
Offset: 0

Views

Author

Seiichi Manyama, May 01 2022

Keywords

Crossrefs

Cf. A121452, A353224.
Cf. A246689, A353223.

Programs

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

Formula

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

A357931 a(n) = Sum_{k=0..floor(n/3)} |Stirling1(n - 2*k,n - 3*k)|.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 7, 13, 27, 57, 120, 262, 593, 1361, 3171, 7559, 18356, 45186, 112927, 286689, 737641, 1921639, 5070154, 13540352, 36566737, 99830013, 275459693, 767798853, 2160953618, 6139721116, 17604534427, 50924095081, 148570523479, 437071675997
Offset: 0

Views

Author

Seiichi Manyama, Oct 21 2022

Keywords

Crossrefs

Cf. A124380, A357932, A357933.
Cf. A353223, A357901, A357925.

Programs

  • Mathematica
    Table[Sum[Abs[StirlingS1[n-2k,n-3k]],{k,0,Floor[n/3]}],{n,0,40}] (* Harvey P. Dale, Nov 01 2023 *)
  • PARI
    a(n) = sum(k=0, n\3, abs(stirling(n-2*k, n-3*k, 1)));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k*prod(j=0, k-1, 1+j*x^2)))

Formula

G.f.: Sum_{k>=0} x^k * Product_{j=0..k-1} (1 + j * x^2).
Showing 1-7 of 7 results.

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

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