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 1-2 of 2 results.

A371304 E.g.f. satisfies A(x) = 1/(1 - x^2*(exp(x*A(x)) - 1)).

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 1470, 12642, 70616, 2131992, 39352410, 470186750, 11032124532, 295053244356, 5896487364950, 146264289411450, 4625791393554480, 130492119237611312, 3837833086814864946, 135471306780659593206, 4800394977109827314060
Offset: 0

Views

Author

Seiichi Manyama, Mar 18 2024

Keywords

Crossrefs

Cf. A370989, A371139.
Cf. A358013, A371302.

Programs

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

Formula

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

A375832 E.g.f. satisfies A(x) = 1/(1 + x*log(1 - x^2*A(x))).

Original entry on oeis.org

1, 0, 0, 6, 0, 60, 1440, 1680, 100800, 1905120, 9979200, 427109760, 8103110400, 102745843200, 3926897694720, 84531807360000, 1844343928627200, 69047821979136000, 1790206583413248000, 54550224714585600000, 2112795340044060672000
Offset: 0

Views

Author

Seiichi Manyama, Aug 30 2024

Keywords

Crossrefs

Cf. A371302.

Programs

  • Mathematica
    Table[n!*Sum[(n-k)!*Abs[StirlingS1[k, n - 2*k]]/(k!*(k + 1)!), {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 31 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\2, (n-k)!*abs(stirling(k, n-2*k, 1))/(k!*(k+1)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (n-k)! * |Stirling1(k,n-2*k)|/( k! * (k+1)! ).
a(n) ~ sqrt(s*(-1 + s*(1 + r^2 + r^2*(-1 + 2*r)*s))/(1 + 2*r*s)) * n^(n-1) / (exp(n) * r^(n+1)), where r = 0.555108855597239653157700556001479889170962... and s = 1.679468515326835651547953595104045902497719... are real roots of the system of equations 1 + r*log(1 - r^2*s) = 1/s, r^3*s^2 = 1 - r^2*s. - Vaclav Kotesovec, Aug 31 2024
Showing 1-2 of 2 results.

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

Content is available under The OEIS End-User License Agreement.