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.

A305824 Number of endofunctions on [n] whose cycle lengths are triangular numbers.

Original entry on oeis.org

1, 1, 3, 18, 157, 1776, 24807, 413344, 8004537, 176630400, 4374300331, 120136735104, 3623854678677, 119102912981248, 4236492477409935, 162152320065532416, 6645233337842716273, 290321208589666369536, 13469914225467040015827, 661442143465113960448000
Offset: 0

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Author

Alois P. Heinz, Jun 10 2018

Keywords

Links

Crossrefs

Cf. A000217, A060435, A116956, A193374, A205799, A273994, A273996, A273998.

Programs

  • Maple
    b:= proc(n) option remember; local r, f, g;
      if n=0 then 1 else r, f, g:=$0..2;
      while f<=n do r, f, g:= r+(f-1)!*
         b(n-f)*binomial(n-1, f-1), f+g, g+1
      od; r fi
    end:
a:= n-> add(b(j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);
  • Mathematica
    b[n_] := b[n] = Module[{r, f, g}, If[n == 0, 1, {r, f, g} = {0, 1, 2}; While[f <= n, {r, f, g} = {r + (f - 1)!*b[n - f]*Binomial[n - 1, f - 1], f + g, g + 1}]; r]];
a[0] = 1; a[n_] := Sum[b[j]*n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 15 2018, after Alois P. Heinz *)

A329256 Expansion of e.g.f. exp(Sum_{k>=1} x^(k^2) / (k^2)!).

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 16, 36, 106, 443, 1796, 6161, 23816, 122266, 643644, 2934296, 14002237, 83835433, 532282819, 3005258539, 17039094646, 115611682810, 848428608644, 5682350940168, 37297365940462, 281594230420802, 2323660209441962, 17929392395804072
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 09 2019

Keywords

Links

Crossrefs

Cf. A000110, A010052, A205799, A205800, A205801, A205802, A205804, A353180.

Programs

  • Mathematica
    nmax = 27; CoefficientList[Series[Exp[Sum[x^(k^2)/(k^2)!, {k, 1, Floor[nmax^(1/2)] + 1}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Boole[IntegerQ[k^(1/2)]] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 27}]
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, sqrtint(N), x^k^2/(k^2)!)))) \\ Seiichi Manyama, Apr 29 2022
  • PARI
    a(n) = if(n==0, 1, sum(k=1, sqrtint(n), binomial(n-1, k^2-1)*a(n-k^2))); \\ Seiichi Manyama, Apr 29 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A010052(k) * a(n-k).

A334385 E.g.f.: Product_{k>=1} (1 + x^(k*(k + 1)/2) / (k*(k + 1)/2)!).

Original entry on oeis.org

1, 1, 0, 1, 4, 0, 1, 7, 0, 84, 841, 11, 0, 286, 4004, 1, 8024, 136136, 816, 7775256, 155195040, 54265, 1193830, 0, 109832360, 2749077760, 84987760, 296010, 10716746041, 310545275069, 1201800600, 2444026056820, 77016647623040, 0, 14402113079955304, 504073957798435640
Offset: 0

Views

Author

Ilya Gutkovskiy, May 11 2020

Keywords

Crossrefs

Cf. A007837, A032310, A053614 (positions of 0's), A115278, A205799.

Programs

  • Mathematica
    nmax = 35; CoefficientList[Series[Product[(1 + x^(k (k + 1)/2)/(k (k + 1)/2)!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[DivisorSum[k, -#/(-#!)^(k/#) &, IntegerQ[Sqrt[8 # + 1]] &] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 35}]
Showing 1-3 of 3 results.

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