OFFSET
0,3
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!.
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
EXAMPLE
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
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 *)
PROG
(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
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
R. H. Hardin, May 23 2011
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Sep 09 2023
STATUS
approved