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 1-5 of 5 results.

A368165 Expansion of e.g.f. -log(1 + x^2/2 * log(1 - x)).

Original entry on oeis.org

0, 0, 0, 3, 6, 20, 180, 1134, 7980, 78840, 798840, 8620920, 107668440, 1449377280, 20755871136, 323448048000, 5398086002400, 95487623038080, 1796842848654720, 35808112038746880, 751616958775939200, 16600116241063514880, 384905905873078867200
Offset: 0

Views

Author

Seiichi Manyama, Dec 14 2023

Keywords

Crossrefs

Programs

  • Mathematica
    With[{nn =30},CoefficientList[Series[-Log[1+x^2/2 Log[1-x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 22 2024 *)
  • PARI
    a(n) = n!*sum(k=1, n\3, (k-1)!*abs(stirling(n-2*k, k, 1))/(2^k*(n-2*k)!));

Formula

a(n) = n! * Sum_{k=1..floor(n/3)} (k-1)! * |Stirling1(n-2*k,k)|/(2^k * (n-2*k)!).
a(0) = a(1) = a(2) = 0; a(n) = n!/(2*(n-2)) + Sum_{k=3..n-1} k!/(2*(k-2)) * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Jan 22 2025

A368166 Expansion of e.g.f. -log(1 + x^3/6 * log(1 - x)).

Original entry on oeis.org

0, 0, 0, 0, 4, 10, 40, 210, 1904, 15120, 132600, 1293600, 14673120, 178738560, 2341182480, 32915282400, 499117301760, 8075042976000, 138689356915200, 2519863488979200, 48354005826489600, 976893364144857600, 20721305503846886400, 460363370406207206400
Offset: 0

Views

Author

Seiichi Manyama, Dec 14 2023

Keywords

Comments

This sequence is different from A351493.

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=1, n\4, (k-1)!*abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!));

Formula

a(n) = n! * Sum_{k=1..floor(n/4)} (k-1)! * |Stirling1(n-3*k,k)|/(6^k * (n-3*k)!).
a(0) = a(1) = a(2) = a(3) = 0; a(n) = n!/(6*(n-3)) + Sum_{k=4..n-1} k!/(6*(k-3)) * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Jan 22 2025

A366752 Expansion of e.g.f. -log(1 + x^2 * log(1 - x)).

Original entry on oeis.org

0, 0, 0, 6, 12, 40, 540, 3528, 25200, 324000, 3648960, 41690880, 622274400, 9573033600, 150465579264, 2705877820800, 51819600268800, 1028013655818240, 22086154437442560, 504188626820659200, 11997041025160396800, 301999960230701322240
Offset: 0

Views

Author

Seiichi Manyama, Dec 15 2023

Keywords

Crossrefs

Programs

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

Formula

a(n) = n! * Sum_{k=1..floor(n/3)} (k-1)! * |Stirling1(n-2*k,k)|/(n-2*k)!.

A366777 Expansion of e.g.f. -log(1 + x^3 * log(1 - x)).

Original entry on oeis.org

0, 0, 0, 0, 24, 60, 240, 1260, 28224, 241920, 2181600, 21621600, 395176320, 5915669760, 87034409280, 1321166246400, 26242709391360, 529649632512000, 10741214992435200, 221702803264051200, 5187617452174233600, 128491776028533657600
Offset: 0

Views

Author

Seiichi Manyama, Dec 15 2023

Keywords

Crossrefs

Programs

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

Formula

a(n) = n! * Sum_{k=1..floor(n/4)} (k-1)! * |Stirling1(n-3*k,k)|/(n-3*k)!.

A380338 Expansion of e.g.f. log(1 - x * log(1 - x)).

Original entry on oeis.org

0, 0, 2, 3, -4, -30, 54, 1260, 3856, -36288, -279000, 2970000, 56725008, 109343520, -5495740992, -26086263840, 1293641890560, 21771049466880, -45508965806592, -4589738336217600, 10493846174810880, 2423866077943511040, 34328754265480012800, -358930542362135546880
Offset: 0

Views

Author

Seiichi Manyama, Jan 21 2025

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(log(1-x*log(1-x)))))
    
  • PARI
    a(n) = n!*sum(k=1, n\2, (-1)^(k-1)*(k-1)!*abs(stirling(n-k, k, 1))/(n-k)!);

Formula

a(n) = n! * Sum_{k=1..floor(n/2)} (-1)^(k-1) * (k-1)! * |Stirling1(n-k,k)|/(n-k)!.
a(0) = a(1) = 0; a(n) = n * (n-2)! - Sum_{k=2..n-1} k * (k-2)! * binomial(n-1,k) * a(n-k).
Showing 1-5 of 5 results.