A215561 -id:A215561 - cp's OEIS frontend

A147681 Late-growing permutations: number of permutations of 1..n with every partial sum <= the same partial sum averaged over all permutations.

1, 1, 1, 3, 7, 35, 139, 1001, 5701, 53109, 402985, 4605271, 43665667, 589809987, 6735960079, 104899483845, 1402547616085, 24698838710457, 378845419610773, 7444522779300351, 128830635114146047, 2792467448952670671, 53854927962971227495, 1276369340371154144337, 27141331409803338993193, 698008560075731437652425, 16228797258964121571885457, 450111715263775132783135875

Offset: 0

R. H. Hardin, May 01 2009

Same as A145874.

David Scambler et al., A147681 Late-growing permutations and follow-up messages on the SeqFan list, Aug 10 2012

This is the first of 19 related sequences, the others being A147682, A147684, A147686, A147687, A147692, A147694, A147695, A147697, A147698, A147700, A147705, A147707, A147712, A147713, A147714, A147715, A147717, A147769.

Column k=1 of A215561.

a:= proc(n) option remember; local b, m; m:= n*(n+1)/2;
      b:= proc(s) option remember; local h, g; h:= nops(s);
            g:= (n-h+1)*(1+n)/2 -m +add(i, i=s); `if`(h<2, 1,
            add(`if`(s[i]<=g, b(subsop(i=NULL, s)), 0), i=1..h))
          end; forget(b);
      b([$1..n])
    end:
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 10 2012

a[n_] := a[n] = Module[{b, m}, m = n*(n+1)/2; b[s_List] := b[s] = Module[{h, g}, h = Length[s]; g = (n-h+1)*(1+n)/2 - m + Total[s]; If[h<2, 1, Sum[If[s[[i]] <= g, b[ReplacePart[s, i -> Sequence[]]], 0], {i, 1, h}]]];  b[Range[n]]]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Mar 13 2015, after Alois P. Heinz *)

a(22) from Alois P. Heinz, Aug 10 2012
a(23) from Alois P. Heinz, Nov 01 2014
a(24)-a(25) from Vaclav Kotesovec, Jan 31 2015
a(26)-a(27) from Vaclav Kotesovec, Sep 07 2016

A007004 a(n) = (3n)! / ((n+1)(n!)^3).

Original entry on oeis.org

1, 3, 30, 420, 6930, 126126, 2450448, 49884120, 1051723530, 22787343150, 504636071940, 11377249621920, 260363981732400, 6034149862347600, 141371511060715200, 3343436236585914480, 79726203788589122490, 1914992149823954412750, 46295775130831740013500

Offset: 0

N. J. A. Sloane, James Propp

nonn

Number of walks with steps (0,1)/North, (1,0)/East and (-1,-1)/Southwest from (0,0) to (0,0) of length 3n, and staying above the line y=x (i.e., any point (x,y) along the walk satisfies y>=x ). - Shanzhen Gao, Nov 09 2010
Number of walks in 3-dimensions using steps (1,0,0), (0,1,0), and (0,0,1) from (0,0,0) to (n,n,n) such that after each step we have y<=x. - Eric Werley, Jun 24 2011
Number of possible necklaces consisting of n white beads, n-1 red beads and n-1 black beads, where two necklaces are considered equivalent if they differ by a cyclic permutation. - Thotsaporn Thanatipanonda, Feb 20 2011

			n=1, three walks: NE(SW), (SW)NE, N(SW)E. - _Shanzhen Gao_, Nov 09 2010

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..150
Alin Bostan, Calcul Formel pour la Combinatoire des Marches [The text is in English], Habilitation à Diriger des Recherches, Laboratoire d'Informatique de Paris Nord, Université Paris 13, December 2017.

Cf. A000108, A060693.

Row n=3 of A215561.

[Factorial(3*n) / ((n+1)*Factorial(n)^3): n in [0..30]]; // Vincenzo Librandi, May 26 2011

seq(binomial(2*n,n)*binomial(3*n,n)/(n+1), n=0..20); # Zerinvary Lajos, May 27 2006

a[n_]:=(3*n)!/((n + 1)*(n!)^3); (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)
CoefficientList[Series[Hypergeometric2F1[1/3,2/3,2,27 x],{x,0,20}],x] (* Harvey P. Dale, Apr 07 2013 *)
Table[Multinomial[n, n, n]/(n + 1), {n, 0, 12}] (* Emanuele Munarini, Oct 25 2016 *)

makelist(multinomial_coeff(n,n,n)/(n+1),n,0,24); /* Emanuele Munarini, Oct 25 2016 */

a(n) = C(2*n,n)*C(3*n,n)/(n+1) = A000108(n)*C(3*n,n). - Zerinvary Lajos, May 27 2006
a(n) = A060693(2n,n) = A088617(2n,n). - Philippe Deléham, Nov 23 2011
a(n) = (3*(3*n-1)*(3*n-2)*a(n-1))/(n*(n+1)) for n>0, a(0)=1. - Alois P. Heinz, Aug 13 2013
a(n) ~ 3^(3*n+1/2)/(2*Pi*n^2). - Vaclav Kotesovec, Sep 06 2016

More terms from Zerinvary Lajos, May 27 2006

A147682 Late-growing permutations: number of permutations of 2 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 1, 2, 30, 403, 18720, 746192, 71892912, 5873837638, 951265850580, 133244998049858, 32484245570649180, 6956417433946216990, 2375465385671586163800, 723157816455776560763294, 329255781245519867317200240, 135189844328107458501296074066, 79079768375837127458516103725820

Offset: 0

R. H. Hardin, May 01 2009

Cf. A147681.

Column k=2 of A215561.

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([2$n]):
seq(a(n), n=1..10);  # Alois P. Heinz, Aug 16 2012

b[l_List] := 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[n_] := b[Table[2, {n}]]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Mar 13 2015, after Alois P. Heinz *)

a(14) from Alois P. Heinz, Aug 16 2012
a(15) from Alois P. Heinz, Nov 02 2014
a(16) from Vaclav Kotesovec, Sep 07 2016
a(17) from Vaclav Kotesovec, Sep 08 2016

A147684 Late-growing permutations: number of permutations of 3 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 1, 5, 420, 40350, 19369350, 9212531290, 13126885205000, 17810026933803520, 55560479543584645500, 164686892656273830526336, 953018107457232657556038400

Offset: 0

R. H. Hardin, May 01 2009

nonn

Cf. A147681.

Column k=3 of A215561.

a(0), a(11) from Alois P. Heinz, Nov 01 2014

A147686 Late-growing permutations: number of permutations of 4 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 14, 6930, 5223915, 27032968200, 164401445439455, 3627155158988429250, 86733224358763671877835, 5469038805616093755410863500

Offset: 1

R. H. Hardin May 01 2009

nonn

Cf. A147681.

Column k=4 of A215561.

A147687 Late-growing permutations: number of permutations of 5 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 42, 126126, 783353872, 44776592395920, 3611684199828856072, 1267664556730792079292048, 551951935901513814954541886968

Offset: 1

R. H. Hardin May 01 2009

nonn

Cf. A147681.

Column k=5 of A215561.

A147692 Late-growing permutations: number of permutations of 6 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 132, 2450448, 129141898872, 82881380383401600, 90695437030756958966384, 515544601327354412382720479328, 4172457328749067883416103335334343279

Offset: 1

R. H. Hardin May 01 2009

nonn

Cf. A147681.

Column k=6 of A215561.

a(8) from Alois P. Heinz, Aug 17 2012

A147694 Late-growing permutations: number of permutations of 7 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 429, 49884120, 22745605840236, 165850226337286576800, 2500267880518574604245088816, 233099041543988273824859604028713600

Offset: 1

R. H. Hardin May 01 2009

nonn

Cf. A147681.

Column k=7 of A215561.

A147695 Late-growing permutations: number of permutations of 8 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 1430, 1051723530, 4206489449301315, 351597937025844947295000, 73839261438738554611424321993670

Offset: 1

R. H. Hardin May 01 2009

nonn

Cf. A147681.

Column k=8 of A215561.

A147697 Late-growing permutations: number of permutations of 9 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Original entry on oeis.org

1, 4862, 22787343150, 807660192541534200, 779279938350147159519336600, 2299118288652572230673921886739695630

Offset: 1

R. H. Hardin, May 01 2009

nonn

Cf. A147681.

Column k=9 of A215561.

cp's OEIS Frontend

A147681 Late-growing permutations: number of permutations of 1..n with every partial sum <= the same partial sum averaged over all permutations.

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A007004 a(n) = (3*n)! / ((n+1)*(n!)^3).

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A147682 Late-growing permutations: number of permutations of 2 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Extensions

A147684 Late-growing permutations: number of permutations of 3 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Views

Author

Keywords

Crossrefs

Extensions

A147686 Late-growing permutations: number of permutations of 4 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

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Author

Keywords

Crossrefs

A147687 Late-growing permutations: number of permutations of 5 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Views

Author

Keywords

Crossrefs

A147692 Late-growing permutations: number of permutations of 6 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Views

Author

Keywords

Crossrefs

Extensions

A147694 Late-growing permutations: number of permutations of 7 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Views

Author

Keywords

Crossrefs

A147695 Late-growing permutations: number of permutations of 8 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

Views

Author

Keywords

Crossrefs

A147697 Late-growing permutations: number of permutations of 9 indistinguishable copies of 1..n with every partial sum <= the same partial sum averaged over all permutations.

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Author

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Crossrefs

A007004 a(n) = (3n)! / ((n+1)(n!)^3).