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.

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A215562 Number of permutations of n indistinguishable copies of 1..4 with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 7, 403, 40350, 5223915, 783353872, 129141898872, 22745605840236, 4206489449301315, 807660192541534200, 159752979289765273698, 32371149339259024610992, 6692030708288364864188400, 1406943391115083641966787200, 300084647544974128326709244080
Offset: 0

Views

Author

Alois P. Heinz, Aug 16 2012

Keywords

Examples

			a(0) = 1: the empty permutation.
a(1) = 7: (1,2,3,4), (1,2,4,3), (1,3,2,4), (1,4,2,3), (2,1,3,4), (2,1,4,3), (2,3,1,4).
a(2) = 403: (1,1,2,2,3,3,4,4), (1,1,2,2,3,4,3,4), ..., (2,3,2,3,1,1,4,4), (2,3,2,3,1,4,1,4).

Links

Crossrefs

Row n=4 of A215561.

Programs

  • Maple
    b:= proc(l) option remember; local m, n, g;
      m, n:= nops(l), add(i, i=l);
      g:= add(i*l[i], i=1..m)-(m+1)/2*(n-1);
     `if`(n<2, 1, add(`if`(l[i]>0 and i<=g,
        b(subsop(i=l[i]-1, l)), 0), i=1..m))
    end:
a:= n-> b([n$4]):
seq(a(n), n=0..15);
  • Mathematica
    b[l_] := b[l] = Module[{m, n, g}, {m, n} = {Length[l], Total[l]}; g = Sum[i*l[[i]], {i, 1, m}] - (m + 1)/2*(n - 1); If[n < 2, 1, Sum[If[l[[i]] > 0 && i <= g, b[ReplacePart[l, i -> l[[i]] - 1]], 0], {i, 1, m}]]];
a[k_] := b[Array[k&, 4]];
a /@ Range[0, 15] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)

Formula

a(n) ~ (phi - sqrt(phi)) * 2^(8*n-1/2) / (Pi^(3/2) * n^(5/2)), where phi = (1+sqrt(5))/2. - Vaclav Kotesovec, Jan 31 2015

A215570 Number of permutations of n indistinguishable copies of 1..5 with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 35, 18720, 19369350, 27032968200, 44776592395920, 82881380383401600, 165850226337286576800, 351597937025844947295000, 779279938350147159519336600, 1789294251011628021153241548800, 4228135363283244543270651711564000, 10232120200642411474243152429724152000
Offset: 0

Views

Author

Alois P. Heinz, Aug 16 2012

Keywords

Examples

			a(1) = 35: (1,2,3,4,5), (1,2,3,5,4), ..., (3,2,1,5,4), (3,2,4,1,5).

Links

Crossrefs

Row n=5 of A215561.

Programs

  • Maple
    b:= proc(l) option remember; local m, n, g;
      m, n:= nops(l), add(i, i=l);
      g:= add(i*l[i], i=1..m)-(m+1)/2*(n-1);
     `if`(n<2, 1, add(`if`(l[i]>0 and i<=g,
        b(subsop(i=l[i]-1, l)), 0), i=1..m))
    end:
a:= n-> b([n$5]):
seq(a(n), n=0..15);
  • Mathematica
    b[l_] := b[l] = Module[{m, n, g}, {m, n} = {Length[l], Total[l]}; g = Sum[i*l[[i]], {i, 1, m}] - (m + 1)/2*(n - 1); If[n < 2, 1, Sum[If[l[[i]] > 0 && i <= g, b[ReplacePart[l, i -> l[[i]] - 1]], 0], {i, 1, m}]]];
a[k_] := b[Array[k&, 5]];
a /@ Range[0, 15] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)

Formula

a(n) ~ (3*sqrt(5)-5) * 5^(5*n) / (8*Pi^2*n^3). - Vaclav Kotesovec, Sep 06 2016
Conjectured recurrence of order 3 and degree 15: 3*(n + 1)*(n + 2)^3*(n + 4)^2*(3*n + 8)*(3*n + 10)*(65*n^3 + 398*n^2 + 781*n + 496)*(n + 3)^4*a(n + 3) - 20*(n + 1)*(n + 2)^3*(5*n + 11)*(5*n + 12)*(5*n + 13)*(5*n + 14)*(910*n^5 + 11032*n^4 + 52047*n^3 + 119686*n^2 + 134365*n + 58980)*(n + 3)^2*a(n + 2) + 25*(n + 1)*(n + 2)*(5*n + 6)*(5*n + 7)*(5*n + 8)*(5*n + 9)*(5*n + 11)*(5*n + 12)*(5*n + 13)*(5*n + 14)*(2015*n^5 + 24428*n^4 + 114387*n^3 + 258294*n^2 + 281088*n + 118368)*a(n + 1) - 250*(5*n + 1)*(5*n + 2)*(5*n + 3)*(5*n + 4)*(5*n + 6)*(5*n + 7)*(5*n + 8)*(5*n + 9)*(5*n + 11)*(5*n + 12)*(5*n + 13)*(5*n + 14)*(65*n^3 + 593*n^2 + 1772*n + 1740)*a(n) = 0. - Manuel Kauers and Christoph Koutschan, Mar 02 2023

A215571 Number of permutations of n indistinguishable copies of 1..6 with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 139, 746192, 9212531290, 164401445439455, 3611684199828856072, 90695437030756958966384, 2500267880518574604245088816, 73839261438738554611424321993670, 2299118288652572230673921886739695630, 74651909129454094524656449803220566122458
Offset: 0

Views

Author

Alois P. Heinz, Aug 16 2012

Keywords

Examples

			a(1) = 139: (1,2,3,4,5,6), (1,2,3,4,6,5), ..., (3,4,2,1,6,5), (3,4,2,5,1,6).

Links

Crossrefs

Row n=6 of A215561.

Programs

  • Maple
    b:= proc(l) option remember; local m, n, g;
      m, n:= nops(l), add(i, i=l);
      g:= add(i*l[i], i=1..m)-(m+1)/2*(n-1);
     `if`(n<2, 1, add(`if`(l[i]>0 and i<=g,
        b(subsop(i=l[i]-1, l)), 0), i=1..m))
    end:
a:= n-> b([n$6]):
seq(a(n), n=0..6);

A215593 Number of permutations of n indistinguishable copies of 1..7 with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 1001, 71892912, 13126885205000, 3627155158988429250, 1267664556730792079292048, 515544601327354412382720479328, 233099041543988273824859604028713600, 113972303622279852972722869873689584148750, 59182016901859077504525075283397206729638923750
Offset: 0

Views

Author

Alois P. Heinz, Aug 16 2012

Keywords

Examples

			a(1) = 1001: (1,2,3,4,5,6,7), (1,2,3,4,5,7,6), ..., (4,3,5,2,1,7,6), (4,3,5,2,6,1,7).

Links

Crossrefs

Row n=7 of A215561.

Programs

  • Maple
    b:= proc(x, y, z, u, v, w, h) option remember; local n, g;
      n:= x+y+z+u+v+w+h; g:= x+2*y+3*z+4*u+5*v+6*w+7*h -8*(n-1)/2;
     `if`(n<2, 1, `if`(x>0 and 1<=g, b(x-1, y, z, u, v, w, h), 0)+
        `if`(y>0 and 2<=g, b(x, y-1, z, u, v, w, h), 0)+
        `if`(z>0 and 3<=g, b(x, y, z-1, u, v, w, h), 0)+
        `if`(u>0 and 4<=g, b(x, y, z, u-1, v, w, h), 0)+
        `if`(v>0 and 5<=g, b(x, y, z, u, v-1, w, h), 0)+
        `if`(w>0 and 6<=g, b(x, y, z, u, v, w-1, h), 0)+
        `if`(h>0 and 7<=g, b(x, y, z, u, v, w, h-1), 0))
    end:
a:= n-> b(n$7):
seq(a(n), n=0..4);
Previous Showing 21-24 of 24 results.

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