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 1-7 of 7 results.

A193374 E.g.f.: A(x) = exp( Sum_{n>=1} x^(n*(n+1)/2) / (n*(n+1)/2) ).

Original entry on oeis.org

1, 1, 1, 3, 9, 21, 201, 1191, 4593, 36009, 620721, 5297931, 40360761, 474989373, 4345942329, 122776895151, 2118941145441, 21344580276561, 303071564084193, 4476037678611219, 59935820004483561, 3838519441659950181, 78361805638079449641, 949279542954821272503
Offset: 0

Views

Author

Paul D. Hanna, Jul 24 2011

Keywords

Comments

Number of permutations of [n] whose cycle lengths are triangular numbers. - Alois P. Heinz, May 12 2016

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 3*x^3/3! + 9*x^4/4! + 21*x^5/5! + 201*x^6/6! +...
where
log(A(x)) = x + x^3/3 + x^6/6 + x^10/10 + x^15/15 + x^21/21 +...

Links

Crossrefs

Cf. A000217, A193375, A273001, A305824, A317130.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(`if`(issqr(8*j+1),
       a(n-j)*(j-1)!*binomial(n-1, j-1), 0), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 12 2016
  • Mathematica
    a[n_] := a[n] = If[n == 0, 1, Sum[If[IntegerQ @ Sqrt[8*j + 1], a[n - j]*(j - 1)!*Binomial[n - 1, j - 1], 0], {j, 1, n}]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 05 2018, after Alois P. Heinz *)
  • PARI
    {a(n)=n!*polcoeff(exp(sum(m=1,sqrtint(2*n+1),x^(m*(m+1)/2)/(m*(m+1)/2)+x*O(x^n))),n)}

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

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.

A317446 Number of permutations of [n] whose lengths of increasing runs are distinct triangular numbers.

Original entry on oeis.org

1, 1, 0, 1, 6, 0, 1, 12, 0, 166, 3687, 20, 0, 570, 18514, 1, 16044, 689458, 1630, 46150176, 2799527248, 108527, 6182180, 0, 653209572, 50529806020, 457774882, 592018, 64091958837, 5934158290988, 7151183666, 15132424235658, 1574449800015044, 0, 342747690810188908
Offset: 0

Views

Author

Alois P. Heinz, Jul 28 2018

Keywords

Links

Crossrefs

Cf. A000217, A053614, A061208, A317130, A317444, A317445, A317447, A317448.

Programs

  • Maple
    g:= (n, s)-> `if`(n in s or not issqr(8*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..40);
  • Mathematica
    g[n_, s_] := If[MemberQ[s, n] || !IntegerQ@Sqrt[8*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, 40}] (* Jean-François Alcover, Jul 14 2021, after Alois P. Heinz *)

Formula

a(n) = 0 <=> n in { A053614 }.
a(n) > 0 <=> n in { A061208 }.

A317165 Number of permutations of [n*(n+1)/2] with distinct lengths of increasing runs.

Original entry on oeis.org

1, 1, 5, 241, 188743, 2734858573, 892173483721887, 7469920269852025033699, 1841449549508718383891930251607, 14973026148724796464136435753195418043885, 4467880642339303169146446437381463615730321314015457, 53810913396105573079543194840166969124601447333276658546225661505
Offset: 0

Views

Author

Alois P. Heinz, Jul 23 2018

Keywords

Crossrefs

Cf. A000217, A246292, A317130, A317166, A317273.

Programs

  • Maple
    g:= (n, s)-> `if`(n in s, 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*(n+1)/2, 0$2, {}):
seq(a(n), n=0..8);
  • Mathematica
    g[n_, s_] := If[MemberQ[s, n], 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, u}], 0] + Sum[b[u + j - 1, o - j, t + 1, s], {j, o}]];
a[n_] := b[n(n+1)/2, 0, 0, {}];
Table[a[n], {n, 0, 8}] (* Jean-François Alcover, Sep 01 2021, after Alois P. Heinz *)

Formula

a(n) = A317166(A000217(n)).
a(n) >= A317273(n).
Showing 1-7 of 7 results.

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