A190917 Number of permutations of n copies of 1..3 introduced in order 1..3 with no element equal to another within a distance of 1.

1, 1, 5, 29, 182, 1198, 8142, 56620, 400598, 2872754, 20824778, 152303410, 1122149800, 8319825040, 62017475600, 464452683432, 3492568119566, 26358270711370, 199565061455634, 1515311001158482, 11535716330003876, 88025068713285476, 673124069796140900

Offset: 0

Author

R. H. Hardin, May 23 2011

Keywords

• nonn

Examples

			All solutions for n=2:
  1    1    1    1    1
  2    2    2    2    2
  3    3    3    3    1
  1    2    2    1    3
  3    3    1    2    2
  2    1    3    3    3
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1111 (terms 1..200 from R. H. Hardin)
• R. J. Mathar, A class of multinomial permutations avoiding object clusters, vixra:1511.0015 (2015), sequence M_{3,m}/3!.

Crossrefs

Column 3 of A322013.

Cf. A000012 (b=2), A190918 (b=4), A190920 (b=5), A190923 (b=6), A190927 (b=7), A190932 (b=8), A321987 (b=9), A322061 (b=10).

Programs

Magma

[(&+[Binomial(n-1, k)*(Binomial(n-1, k)*Binomial(2*n+1-2*k, n+1) + Binomial(n-1, k+1)*Binomial(2*n-2*k, n+1)): k in [0..Floor(n/2)]])/3: n in [1..25]]; // G. C. Greubel, Nov 24 2018

Maple

a:= proc(n) option remember; `if`(n<3, [1$2, 5][n+1],
      ((7*n-4)*a(n-1)+8*(n-2)^2*a(n-2)/(n+1))/n)
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Sep 09 2023

Mathematica

Table[(1/3)*Sum[Binomial[n-1, k]*(Binomial[n-1, k]*Binomial[2*n+1-2*k, n+1] + Binomial[n-1, k+1]*Binomial[2*n-2*k, n+1]), {k,0,Floor[n/2]}], {n,1,25}] (* G. C. Greubel, Nov 24 2018 *)

PARI

A190917(n) = sum(k=0, n\2, binomial(n-1, k)*(binomial(n-1, k)*binomial(2*n+1-2*k, n+1)+binomial(n-1, k+1)*binomial(2*n-2*k, n+1))) / 3; \\ Max Alekseyev, Dec 10 2017

Sage

[(1/3)*sum(binomial(n-1, k)*(binomial(n-1, k)*binomial(2*n+1-2*k, n+1) + binomial(n-1, k+1)*binomial(2*n-2*k, n+1))  for k in range(1+floor(n/2))) for n in (1..25)] # G. C. Greubel, Nov 24 2018

Formula

a(n) = A110706(n) / 6 for n >= 1.

n*(n+1)*a(n) - (n+1)*(7*n-4)*a(n-1) - 8*(n-2)^2*a(n-2) = 0. - R. J. Mathar, Nov 01 2015 from A110706

Extensions

a(0)=1 prepended by Alois P. Heinz, Sep 09 2023