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 1-6 of 6 results.

A371121 E.g.f. satisfies A(x) = 1 - x*A(x)*log(1 - x*A(x)).

Original entry on oeis.org

1, 0, 2, 3, 56, 330, 5724, 68460, 1351552, 24594192, 578257200, 13915923120, 389216689344, 11518744311360, 377576873670528, 13185760854520800, 497969104450867200, 19992393239486976000, 856421361373185137664, 38819358713756193292800
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = (n!)^2 * Sum_{k=0..floor(n/2)} |Stirling1(n-k,k)|/( (n-k)! * (n-k+1)! ).
E.g.f.: (1/x) * Series_Reversion( x/(1 - x*log(1 - x)) ). - Seiichi Manyama, Sep 19 2024

A371139 E.g.f. satisfies A(x) = 1 + x^2*A(x)^2*(exp(x*A(x)) - 1).

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 2190, 17682, 94136, 4762872, 83210490, 920248670, 34266719652, 948535937076, 17568958623398, 607198057666410, 22018456385103600, 595499717140604912, 21682086461493768306, 926586132659265073590, 33197900968981072951580
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = (n!)^2 * Sum_{k=0..floor(n/3)} Stirling2(n-2*k,k)/( (n-2*k)! * (n-k+1)! ).
E.g.f.: (1/x) * Series_Reversion( x/(1 + x^2*(exp(x) - 1)) ). - Seiichi Manyama, Sep 19 2024

A371120 E.g.f. satisfies A(x) = 1 + x*A(x)^3*(exp(x*A(x)) - 1).

Original entry on oeis.org

1, 0, 2, 3, 100, 545, 17946, 203497, 7194440, 132963777, 5172409630, 135827977241, 5868623306844, 200952952956769, 9665278822378466, 407661518051710665, 21789972653746494736, 1088515671895571005313, 64406426353877958253254
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2024

Keywords

Crossrefs

Programs

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

Formula

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

A376293 E.g.f. satisfies A(x) = 1 + (x*A(x))^3 * (exp(x*A(x)) - 1).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 120, 210, 161616, 1633464, 10584720, 54886590, 10785520680, 243865703796, 3309354530664, 34340235932730, 3229131046905120, 123251776925401200, 2846181122195004576, 49221175229381943414, 3060186440577720774840
Offset: 0

Views

Author

Seiichi Manyama, Sep 19 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!^2*sum(k=0, n\4, stirling(n-3*k, k, 2)/((n-3*k)!*(n-k+1)!));

Formula

a(n) = (n!)^2 * Sum_{k=0..floor(n/4)} Stirling2(n-3*k,k)/( (n-3*k)! * (n-k+1)! ).
E.g.f.: (1/x) * Series_Reversion( x/(1 + x^3*(exp(x) - 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)! ).

A377393 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*(exp(x) - 1))^3 ).

Original entry on oeis.org

1, 0, 6, 9, 516, 3075, 149418, 1956171, 95139432, 2099836899, 108189172830, 3465051871083, 194015893087404, 8207832658120563, 505114926236953074, 26525536061251639275, 1800555184934893332048, 112493970299385975997635, 8415880480577316204054630
Offset: 0

Views

Author

Seiichi Manyama, Oct 27 2024

Keywords

Crossrefs

Programs

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

Formula

E.g.f. satisfies A(x) = ( 1 + x*A(x) * (exp(x*A(x)) - 1) )^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A371272.
a(n) = 3 * n! * (3*n+2)! * Sum_{k=0..floor(n/2)} Stirling2(n-k,k)/( (n-k)! * (3*n-k+3)! ).
Showing 1-6 of 6 results.