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 1-6 of 6 results.

A317130 Number of permutations of [n] whose lengths of increasing runs are triangular numbers.

Original entry on oeis.org

1, 1, 1, 2, 7, 24, 93, 483, 2832, 17515, 123226, 978405, 8312802, 75966887, 756376739, 8070649675, 91320842018, 1099612368110, 14054043139523, 189320856378432, 2682416347625463, 39945105092501742, 623240458310527252, 10160826473676346731, 172871969109661492526
Offset: 0

Views

Author

Alois P. Heinz, Jul 21 2018

Keywords

Examples

			a(2) = 1: 21.
a(3) = 2: 123, 321.
a(4) = 7: 1243, 1342, 2134, 2341, 3124, 4123, 4321.
a(5) = 24: 12543, 13542, 14532, 21354, 21453, 23541, 24531, 31254, 31452, 32145, 32451, 34521, 41253, 41352, 42135, 42351, 43125, 51243, 51342, 52134, 52341, 53124, 54123, 54321.

Links

Crossrefs

Cf. A000217, A097597, A193374, A317111, A317128, A317129, A317131, A317132, A317446.

Programs

  • Maple
    g:= n-> `if`(issqr(8*n+1), 1, 0):
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-> b(n, 0$2):
seq(a(n), n=0..27);
  • Mathematica
    g[n_] := If[IntegerQ @ Sqrt[8n+1], 1, 0];
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_] := b[n, 0, 0];
a /@ Range[0, 27] (* Jean-François Alcover, Apr 29 2020, after Alois P. Heinz *)

A317128 Number of permutations of [n] whose lengths of increasing runs are Fibonacci numbers.

Original entry on oeis.org

1, 1, 2, 6, 23, 112, 652, 4425, 34358, 299971, 2910304, 31059715, 361603228, 4560742758, 61947243329, 901511878198, 13994262184718, 230811430415207, 4030772161073249, 74301962970014978, 1441745847111969415, 29374226224980834077, 626971133730275593916
Offset: 0

Views

Author

Alois P. Heinz, Jul 21 2018

Keywords

Links

Crossrefs

Cf. A000045, A097597, A273001, A317111, A317129, A317130, A317131, A317132, A317444.

Programs

  • Maple
    g:= n-> (t-> `if`(issqr(t+4) or issqr(t-4), 1, 0))(5*n^2):
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-> b(n, 0$2):
seq(a(n), n=0..27);
  • Mathematica
    g[n_] := With[{t = 5n^2}, If[IntegerQ@Sqrt[t+4] || IntegerQ@Sqrt[t-4], 1, 0]];
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_] := b[n, 0, 0];
a /@ Range[0, 27] (* Jean-François Alcover, Mar 29 2021, after Alois P. Heinz *)

A317129 Number of permutations of [n] whose lengths of increasing runs are squares.

Original entry on oeis.org

1, 1, 1, 1, 2, 9, 40, 151, 571, 2897, 19730, 140190, 953064, 6708323, 54631552, 510143776, 4987278692, 49168919669, 505209884549, 5638095015594, 67921924172174, 852861260421398, 10992380368532792, 147296144926635359, 2082906807168675698, 30973237281668975230
Offset: 0

Views

Author

Alois P. Heinz, Jul 21 2018

Keywords

Examples

			a(3) = 1: 321.
a(4) = 2: 1234, 4321.
a(5) = 9: 12354, 12453, 13452, 21345, 23451, 31245, 41235, 51234, 54321.

Links

Crossrefs

Cf. A000290, A097597, A205801, A317111, A317128, A317130, A317131, A317132, A317445.

Programs

  • Maple
    g:= n-> `if`(issqr(n), 1, 0):
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-> b(n, 0$2):
seq(a(n), n=0..27);
  • Mathematica
    g[n_] := If[IntegerQ@Sqrt[n], 1, 0];
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_] := b[n, 0, 0];
a /@ Range[0, 27] (* Jean-François Alcover, Mar 29 2021, after Alois P. Heinz *)

A317131 Number of permutations of [n] whose lengths of increasing runs are prime numbers.

Original entry on oeis.org

1, 0, 1, 1, 5, 19, 80, 520, 2898, 22486, 171460, 1509534, 14446457, 147241144, 1650934446, 19494460567, 248182635904, 3340565727176, 47659710452780, 718389090777485, 11381176852445592, 189580213656445309, 3305258537062221020, 60273557241570401742
Offset: 0

Views

Author

Alois P. Heinz, Jul 21 2018

Keywords

Examples

			a(2) = 1: 12.
a(3) = 1: 123.
a(4) = 5: 1324, 1423, 2314, 2413, 3412.
a(5) = 19: 12345, 12435, 12534, 13245, 13425, 13524, 14235, 14523, 15234, 23145, 23415, 23514, 24135, 24513, 25134, 34125, 34512, 35124, 45123.

Links

Crossrefs

Cf. A000040, A097597, A218002, A317111, A317128, A317129, A317130, A317132, A317447.

Programs

  • Maple
    g:= n-> `if`(n=0 or isprime(n), 1, 0):
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-> b(n, 0$2):
seq(a(n), n=0..27);
  • Mathematica
    g[n_] := If[n == 0 || PrimeQ[n], 1, 0];
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_] := b[n, 0, 0];
a /@ Range[0, 27] (* Jean-François Alcover, Mar 29 2021, after Alois P. Heinz *)
  • Python
    from functools import lru_cache
from sympy import isprime
def g(n): return int(n == 0 or isprime(n))
@lru_cache(maxsize=None)
def b(u, o, t):
  if u + o == 0: return g(t)
  return (sum(b(u-j,  o+j-1,  1) for j in range(1, u+1)) if g(t) else 0) +\
          sum(b(u+j-1, o-j, t+1) for j in range(1, o+1))
def a(n): return b(n, 0, 0)
print([a(n) for n in range(28)]) # Michael S. Branicky, Mar 29 2021 after Alois P. Heinz

A317448 Number of permutations of [n] whose lengths of increasing runs are distinct factorial numbers.

Original entry on oeis.org

1, 1, 1, 4, 0, 0, 1, 12, 54, 1002, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 48, 648, 39444, 0, 0, 1187548, 96978608, 1721374454, 169149221140, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Alois P. Heinz, Jul 28 2018

Keywords

Links

Crossrefs

Cf. A000142, A059590, A115945, A317132, A317444, A317445, A317446, A317447.

Programs

  • Maple
    h:= proc(n) local i; 1; for i from 2 do
      if n=% then 1; break elif n<% then 0; break fi;
      %*i od; h(n):=%
    end:
g:= (n, s)-> `if`(n in s or not (n=0 or h(n)=1), 0, 1):
b:= proc(u, o, t, s) option remember; `if`(u+o=0, g(t, s),
      `if`(g(t, s)=1, add(b(u-j, o+j-1, 1, s union {t})
       , j=1..u), 0)+ add(b(u+j-1, o-j, t+1, s), j=1..o))
    end:
a:= n-> b(n, 0$2, {}):
seq(a(n), n=0..34);
  • Mathematica
    h[n_] := Module[{i, pc = 1}, For[i = 2, True, i++, Which[n == pc, pc = 1; Break[], n < pc, pc = 0; Break[]]; pc = pc*i]; h[n] = pc];
g[n_, s_] := If[MemberQ[s, n] || !(n == 0 || h[n] == 1), 0, 1];
b[u_, o_, t_, s_] := b[u, o, t, s] = If[u + o == 0, g[t, s],
     If[g[t, s] == 1, Sum[b[u - j, o + j - 1, 1, s ~Union~ {t}],
     {j, 1, u}], 0] + Sum[b[u + j - 1, o - j, t + 1, s], {j, 1, o}]];
a[n_] := b[n, 0, 0, {}];
Table[a[n], {n, 0, 34}] (* Jean-François Alcover, Jul 14 2021, after Alois P. Heinz *)

Formula

a(n) = 0 <=> n in { A115945 }.
a(n) > 0 <=> n in { A059590 }.

A272603 Number of permutations of [n] whose cycle lengths are factorials.

Original entry on oeis.org

1, 1, 2, 4, 10, 26, 196, 1072, 7484, 42940, 261496, 1477136, 15219832, 134828344, 1488515120, 13692017536, 130252442896, 1123580329232, 14639510308384, 173489066401600, 2528654220104096, 31472160333513376, 402634734214583872, 4645625988351336704, 25925035549644280991680
Offset: 0

Views

Author

Joerg Arndt, May 29 2016

Keywords

Links

Crossrefs

Cf. A000142, A273001 (cycle lengths are Fibonacci numbers), A272602 (e.g.f.: exp( sum(n>=1, x^(n!) / n ) ) ), A273996, A317132.

Programs

  • Maple
    a:= proc(n) option remember; local r, f, i;
      if n=0 then 1 else r, f, i:= $0..2;
        while f<=n do r:= r +a(n-f)*(f-1)!*
          binomial(n-1, f-1); f, i:= f*i, i+1
        od; r
      fi
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 04 2016
  • Mathematica
    nmax = 4; egf = Exp[Sum[x^n!/n!, {n, 1, nmax}]] + O[x]^(nmax! + 1); CoefficientList[egf, x]*Range[0, nmax!]! (* Jean-François Alcover, Feb 19 2017 *)
  • 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 1-6 of 6 results.

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