OFFSET
1,1
LINKS
R. H. Hardin, Table of n, a(n) for n = 1..1475
FORMULA
Empirical for row n:
n=1: a(k)=2*a(k-1)-a(k-2)
n=2: a(k)=2*a(k-1)-2*a(k-3)+a(k-4)
n=3: (order 12 antisymmetric)
n=4: (order 32 symmetric)
n=5: (order 84 symmetric)
T(n, k) = Sum_{s=0..k} Product_{L=2..n} NC(s*L, L, k), where NC(s, n, k) is the number of compositions of sum s with n parts between 0 and k. - M. F. Hasler, Aug 21 2025
EXAMPLE
Table starts
.2.....3......4.......5........6.........7.........8..........9.........10
.2.....5......8......13.......18........25........32.........41.........50
.2....23.....62.....157......312.......601.......986.......1619.......2426
.2...401...1862...10177....33352....103651....250042.....589763....1199614
.2.20351.187862.3330677.20608352.121537201.493575042.1877543213.5767190924
Some solutions for n = k = 4:
.....2........1........2........2........2........2........2........2
....3.1......0.2......2.2......3.1......2.2......1.3......4.0......4.0
...3.2.1....0.3.0....3.2.1....2.4.0....0.2.4....3.0.3....1.2.3....4.0.2
..2.2.3.1..2.1.0.1..1.2.4.1..4.2.2.0..1.4.3.0..4.0.2.2..3.2.3.0..4.0.4.0
PROG
(PARI) /* helper function mult() gives multiplicity of a composition */
mult(p, L=1, m=(#p)!)={for(k=2, #p, p[k]!=p[k-1] && m\=(-L+L=k)!); m\(#p-L+1)!}
A214595(n, k)={sum(a=1, k, prod(L=2, n, my(c=0); forpart(p=L*a, c+=mult(p), [0, k], L); c))+1} \\ M. F. Hasler, Aug 21 2025
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin, Jul 22 2012
STATUS
approved
