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 1-4 of 4 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

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

A357932 a(n) = Sum_{k=0..floor(n/4)} |Stirling1(n - 3*k,n - 4*k)|.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 7, 11, 18, 33, 64, 122, 227, 428, 838, 1684, 3396, 6841, 13912, 28787, 60398, 127559, 270687, 579055, 1251706, 2730345, 5994501, 13238058, 29436628, 65951104, 148777927, 337606123, 770418129, 1768566987, 4084504483, 9486890220
Offset: 0

Views

Author

Seiichi Manyama, Oct 21 2022

Keywords

Crossrefs

Cf. A124380, A357931, A357933.
Cf. A353225, A357902, A357926.

Programs

  • PARI
    a(n) = sum(k=0, n\4, abs(stirling(n-3*k, n-4*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^3)))

Formula

G.f.: Sum_{k>=0} x^k * Product_{j=0..k-1} (1 + j * x^3).

A357965 Expansion of e.g.f. exp( (exp(x^4) - 1)/x^3 ).

Original entry on oeis.org

1, 1, 1, 1, 1, 61, 361, 1261, 3361, 68041, 1073521, 8343721, 43290721, 432509221, 11472541081, 165124339381, 1457296102081, 12237047593681, 322364521392481, 7462073325643921, 103362225413048641, 1051987428484484941, 21127644716862970441
Offset: 0

Views

Author

Seiichi Manyama, Oct 22 2022

Keywords

Crossrefs

Cf. A357962, A357964.
Cf. A353225.

Programs

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

Formula

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

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