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 1-3 of 3 results.

A381022 Expansion of e.g.f. -log(1-x)^3 * exp(x) / 6.

Original entry on oeis.org

0, 0, 0, 1, 10, 75, 545, 4179, 34860, 318926, 3197210, 34975061, 415371726, 5328246417, 73470506291, 1084206640399, 17054915985752, 284945098917980, 5040033650314996, 94099409345964169, 1849525745917903666, 38176559589575462327, 825716052360614856485, 18675737859143938658251
Offset: 0

Views

Author

Seiichi Manyama, Feb 12 2025

Keywords

Crossrefs

Column k=3 of A094816.
Cf. A381024.

Programs

  • Mathematica
    nmax=23; CoefficientList[Series[-Log[1-x]^3*Exp[x]/6, {x, 0, nmax}], x]Range[0, nmax]! (* Stefano Spezia, Feb 12 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n, k)*abs(stirling(k, 3, 1)));

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * |Stirling1(k,3)|.

A381067 Expansion of e.g.f. log(1-x)^2 * exp(-x) / (2 * (1-x)).

Original entry on oeis.org

0, 0, 1, 3, 17, 100, 694, 5453, 48082, 470328, 5057331, 59313287, 753695139, 10316991100, 151373235896, 2370151632977, 39450142911652, 695612154233648, 12953591498092101, 254044853932550091, 5234026736314790581, 113025076301648693844, 2552830193825115461786
Offset: 0

Views

Author

Seiichi Manyama, Feb 12 2025

Keywords

Crossrefs

Column k=3 of A269954 (with a different offset).

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*abs(stirling(k+1, 3, 1)));

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * |Stirling1(k+1,3)|.
a(n) = A381065(n) + A381065(n+1).

A381108 Expansion of e.g.f. log(1-x)^2 * (exp(x) - 1) / (2 * (1-x)).

Original entry on oeis.org

0, 0, 0, 3, 30, 245, 2010, 17549, 165942, 1705584, 19024275, 229478689, 2981315139, 41545542818, 618579336284, 9804891730633, 164897938095108, 2933486106772376, 55047126101826453, 1086816606230786217, 22523274090016854661, 488907589907823010158, 11093875133012393113766
Offset: 0

Views

Author

Seiichi Manyama, Feb 14 2025

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=0, n-1, binomial(n, k)*abs(stirling(k+1, 3, 1)));

Formula

a(n) = Sum_{k=0..n-1} binomial(n,k) * |Stirling1(k+1,3)|.
a(n) = A381024(n) - A000399(n+1).
Showing 1-3 of 3 results.