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 1-4 of 4 results.

A273001 Number of permutations of [n] whose cycle lengths are Fibonacci numbers.

Original entry on oeis.org

1, 1, 2, 6, 18, 90, 420, 2220, 19020, 130860, 1096920, 9862920, 83843640, 1411202520, 16144792560, 203091829200, 2989264122000, 37012939750800, 597962683188000, 8681244913692000, 126467701221607200, 5006833609034743200, 95602098255580238400
Offset: 0

Views

Author

Alois P. Heinz, May 12 2016

Keywords

Links

Crossrefs

Cf. A000045, A193374, A205801, A218002, A272603, A273994, A317128.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(issqr(5*j^2+4) or issqr(5*j^2-4),
       a(n-j)*(j-1)!*binomial(n-1, j-1), 0), j=1..n))
    end:
seq(a(n), n=0..25);
  • Mathematica
    a[n_] := a[n] = If[n == 0, 1, Sum[If[IntegerQ @ Sqrt[5*j^2+4] || IntegerQ @ Sqrt[5*j^2-4], a[n-j]*(j-1)!*Binomial[n-1, j-1], 0], {j, 1, n}]]; Table[ a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 30 2017, translated from Maple *)

Formula

E.g.f.: exp(Sum_{n>=2} x^F(n)/F(n)) with F = A000045.

A317132 Number of permutations of [n] whose lengths of increasing runs are factorials.

Original entry on oeis.org

1, 1, 2, 5, 17, 70, 350, 2029, 13495, 100813, 837647, 7652306, 76282541, 823684964, 9578815164, 119346454671, 1586149739684, 22397700381817, 334879465463998, 5285103821004717, 87800206978975107, 1531533620821692217, 27987305231654121046, 534688325008397289484
Offset: 0

Views

Author

Alois P. Heinz, Jul 21 2018

Keywords

Links

Crossrefs

Cf. A000142, A097597, A272603, A317111, A317128, A317129, A317130, A317131, A317448.

Programs

  • Maple
    g:= proc(n) local i; 1; for i from 2 do
      if n=% then 1; break elif n<% then 0; break fi;
      %*i od; g(n):=%
    end:
b:= proc(u, o, t) option remember; `if`(u+o=0, g(t),
      `if`(g(t)=1, add(b(u-j, o+j-1, 1), j=1..u), 0)+
       add(b(u+j-1, o-j, t+1), j=1..o))
    end:
a:= n-> `if`(n=0, 1, add(b(j-1, n-j, 1), j=1..n)):
seq(a(n), n=0..27);
  • Mathematica
    g[n_] := g[n] = Module[{i, k = 1}, For[i = 2, True, i++,
     If[n == k, k = 1; Break[]]; If[n < k, k = 0; Break[]];
     k = k*i]; k];
b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, g[t],
     If[g[t] == 1, Sum[b[u - j, o + j - 1, 1], {j, 1, u}], 0] +
     Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}]];
a[n_] := If[n == 0, 1, Sum[b[j - 1, n - j, 1], {j, 1, n}]];
a /@ Range[0, 27] (* Jean-François Alcover, Mar 29 2021~, after Alois P. Heinz *)

Formula

a(3) = 5: 132, 213, 231, 312, 321.
a(4) = 17: 1324, 1423, 1432, 2143, 2314, 2413, 2431, 3142, 3214, 3241, 3412, 3421, 4132, 4213, 4231, 4312, 4321.

A273996 Number of endofunctions on [n] whose cycle lengths are factorials.

Original entry on oeis.org

1, 1, 4, 25, 218, 2451, 33952, 560407, 10750140, 235118665, 5775676496, 157448312649, 4716609543736, 154007821275595, 5443783515005760, 207093963680817511, 8436365861409555728, 366403740283162634193, 16900793597898691865920, 825115046704241167668025
Offset: 0

Views

Author

Alois P. Heinz, Jun 06 2016

Keywords

Links

Crossrefs

Cf. A000142, A060435, A116956, A272603, A273994, A273997, A273998, A305824.

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:= r+(f-1)!*b(n-f)*
         binomial(n-1, f-1); f, g:= 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 = r + (f - 1)!*b[n - f]*Binomial[n - 1, f - 1]; {f, g} = {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 06 2018, from Maple *)

A272602 E.g.f.: exp( sum(n>=1, x^(n!) / n ) ).

Original entry on oeis.org

1, 1, 2, 4, 10, 26, 316, 1912, 14204, 83260, 513496, 2918576, 43605112, 442060984, 5396258960, 51589877536, 502620599056, 4387660231952, 77691034349344, 1065687761143360, 17601265218632096, 230878891157853856, 3055183837594858432, 35865234134641395584, 155845649043266241280960
Offset: 0

Views

Author

Joerg Arndt, May 29 2016

Keywords

Links

Crossrefs

Cf. A272603 (e.g.f.: exp( sum(n>=1, x^(n!) / n! ) ) ).

Programs

  • PARI
    N=66;x='x+O('x^N); Vec(serlaplace(exp(sum(n=1,10,x^(n!)/n))))

Formula

E.g.f.: exp( sum(n>=1, x^(n!) / n ) ).
Showing 1-4 of 4 results.

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