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.

A371262 E.g.f. satisfies A(x) = 1 + x * (exp(x*A(x)^2) - 1).

Original entry on oeis.org

1, 0, 2, 3, 52, 365, 5286, 76867, 1341320, 27823833, 624467530, 16163482511, 452003629452, 13975370745349, 467133121195118, 16865722845267675, 653859200911607056, 27061461284541490097, 1192488605596282310802, 55686113074253206544167
Offset: 0

Views

Author

Seiichi Manyama, Mar 16 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, (2*n-2*k)!*stirling(n-k, k, 2)/((n-k)!*(2*n-3*k+1)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (2*n-2*k)! * Stirling2(n-k,k)/( (n-k)! * (2*n-3*k+1)! ).

A371271 E.g.f. satisfies A(x) = 1 + x*A(x)^3 * (exp(x*A(x)^2) - 1).

Original entry on oeis.org

1, 0, 2, 3, 124, 725, 28146, 352807, 14395256, 298559529, 13269150190, 394087597211, 19361289265044, 752705527798237, 41083484117561354, 1970818974867113295, 119467697774656366576, 6792102349650727753553, 455778318504089893121766
Offset: 0

Views

Author

Seiichi Manyama, Mar 16 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, (2*n+k)!*stirling(n-k, k, 2)/(n-k)!)/(2*n+1)!;

Formula

a(n) = (n!/(2*n+1)!) * Sum_{k=0..floor(n/2)} (2*n+k)! * Stirling2(n-k,k)/(n-k)!.

A371269 E.g.f. satisfies A(x) = 1 + x*A(x) * (exp(x*A(x)^2) - 1).

Original entry on oeis.org

1, 0, 2, 3, 76, 485, 10746, 146167, 3552312, 75642345, 2150551990, 61400333291, 2061654862356, 72804918721405, 2858153637295698, 119363732105632575, 5395737275060765296, 259270058379207421649, 13294348104095211012462, 721446934706871966578899
Offset: 0

Views

Author

Seiichi Manyama, Mar 16 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, (2*n-k)!*stirling(n-k, k, 2)/((n-k)!*(2*n-2*k+1)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (2*n-k)! * Stirling2(n-k,k)/( (n-k)! * (2*n-2*k+1)! ).

A377392 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*(exp(x) - 1))^2 ).

Original entry on oeis.org

1, 0, 4, 6, 224, 1330, 42912, 548114, 18337440, 382829346, 14098368080, 413342914402, 17124811116624, 644015140354898, 30163665817167456, 1375047846420311730, 72583022771706823232, 3866142693873431519554, 228486372085027819754928, 13871056133441358772777154
Offset: 0

Views

Author

Seiichi Manyama, Oct 27 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = 2*n!*(2*n+1)!*sum(k=0, n\2, stirling(n-k, k, 2)/((n-k)!*(2*n-k+2)!));

Formula

E.g.f. satisfies A(x) = ( 1 + x*A(x) * (exp(x*A(x)) - 1) )^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A371270.
a(n) = 2 * n! * (2*n+1)! * Sum_{k=0..floor(n/2)} Stirling2(n-k,k)/( (n-k)! * (2*n-k+2)! ).
Showing 1-4 of 4 results.