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.

A353223 Expansion of e.g.f. (1 - x^3)^(-1/x^2).

Original entry on oeis.org

1, 1, 1, 1, 13, 61, 181, 2101, 19321, 107353, 1338121, 18021961, 153519301, 2162889301, 37434929533, 437750929981, 7054260835441, 146656527486001, 2197288472426641, 40414798347009553, 970905798377330941, 17791752518018762221, 370864149434372540101
Offset: 0

Views

Author

Seiichi Manyama, May 01 2022

Keywords

Crossrefs

Cf. A121452, A246689, A353222, A353225.

Programs

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

Formula

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

A353226 Expansion of e.g.f. (1 - x^2)^(-x).

Original entry on oeis.org

1, 0, 0, 6, 0, 60, 360, 1680, 20160, 151200, 1663200, 17962560, 219542400, 2854051200, 40441040640, 606356150400, 9793028044800, 166481476761600, 3017626733721600, 57359043873331200, 1153275200453376000, 24233844054131712000, 535361100608439705600
Offset: 0

Views

Author

Seiichi Manyama, May 01 2022

Keywords

Crossrefs

Cf. A066166, A121452, A351155, A353227.

Programs

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

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=2..floor((n+1)/2)} (2*k-1)/(k-1) * a(n-2*k+1)/(n-2*k+1)!.
a(n) = n! * Sum_{k=0..floor(n/2)} |Stirling1(k,n-2*k)|/k!.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (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

A357962 Expansion of e.g.f. exp( (exp(x^2) - 1)/x ).

Original entry on oeis.org

1, 1, 1, 4, 13, 51, 271, 1366, 8849, 58717, 432541, 3530176, 29787781, 279974839, 2715912291, 28415168146, 312503079841, 3600714035321, 43979791574809, 556150585730140, 7417561518005341, 102438949373356891, 1476634705941320311, 22102618328057267694
Offset: 0

Views

Author

Seiichi Manyama, Oct 22 2022

Keywords

Crossrefs

Cf. A357964, A357965.
Cf. A121452, A357966.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[(Exp[x^2]-1)/x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 19 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((exp(x^2)-1)/x)))
  • PARI
    a(n) = n!*sum(k=0, n\2, stirling(n-k, n-2*k, 2)/(n-k)!);

Formula

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

A375798 Expansion of e.g.f. 1/(1 + (log(1 - x^2))/x).

Original entry on oeis.org

1, 1, 2, 9, 48, 340, 2820, 27720, 309120, 3897936, 54472320, 838918080, 14080651200, 256214724480, 5018771197440, 105361754097600, 2358985057228800, 56124276848640000, 1413738138502609920, 37591686093776855040, 1052149579611011481600
Offset: 0

Views

Author

Seiichi Manyama, Aug 29 2024

Keywords

Crossrefs

Cf. A007840, A375799.
Cf. A121452.

Programs

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

Formula

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

A353224 Expansion of e.g.f. (1 - x^4)^(-1/x).

Original entry on oeis.org

1, 0, 0, 6, 0, 0, 360, 2520, 0, 60480, 1814400, 13305600, 19958400, 1556755200, 39956716800, 337815878400, 1743565824000, 103742166528000, 2676547896422400, 26863293006950400, 287217598187520000, 15976056520359936000, 432428057769996288000
Offset: 0

Views

Author

Seiichi Manyama, May 01 2022

Keywords

Crossrefs

Cf. A121452, A353222.
Cf. A353225.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^4)^(-1/x)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-log(1-x^4)/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)\4, (4*j-1)/j*v[i-4*j+2]/(i-4*j+1)!)); v;

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..floor((n+1)/4)} (4*k-1)/k * a(n-4*k+1)/(n-4*k+1)!.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (4*exp(n)). - Vaclav Kotesovec, May 04 2022
Showing 1-7 of 7 results.

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