OFFSET
1,2
COMMENTS
The value is smaller than the number of compositions of n into k parts and at least the number of (unordered) partitions.
It is also at least the number of compositions of n into n parts divided by n. From these bounds: C(2*n-1,n-1)/n <= a(n) <= C(2*n-1,n-1). - Robert Gerbicz, Apr 06 2011
a(n) is also the number of Dyck paths of semilength 2n such that each level has exactly n peaks or no peaks. a(3) = 4: //\\//\\//\\, ///\\//\/\\\, ///\/\\//\\\, ////\/\/\\\\. - Alois P. Heinz, Jun 04 2017
LINKS
Robert Gerbicz, Table of n, a(n) for n = 1..500
EXAMPLE
a(4) = 11: the 11 compositions of this type of 4 into 4 parts being
(4,0,0,0); (3,1,0,0); (3,0,1,0); (3,0,0,1);
(2,2,0,0); (2,0,2,0); (2,0,0,2); (2,1,1,0);
(2,1,0,1); (2,0,1,1); (1,1,1,1)
MAPLE
b:= proc(n, i, m) option remember; if n<0 then 0 elif n=0 then 1 elif i=1 then `if`(n<=m, 1, 0) else add(b(n-k, i-1, m), k=0..m) fi end: A:= (n, k)-> add(b(n-m, k-1, m), m=ceil(n/k)..n): seq(A(n, n), n=1..30); # Alois P. Heinz, Jun 14 2009
MATHEMATICA
b[n_, i_, m_] := b[n, i, m] = Which[n<0, 0, n==0, 1, i==1, If[n <= m, 1, 0], True, Sum[b[n-k, i-1, m], {k, 0, m}]]; A[n_, k_] := Sum[b[n-m, k-1, m], {m, Ceiling[n/k], n}]; Table[A[n, n], {n, 1, 30}] (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)
PROG
(PARI) N=120; v=vector(N, i, 0); for(d=1, N, A=matrix(N, N, i, j, 0); A[1, 1]=1; for(i=1, N-1, for(j=0, N-1, s=0; for(k=0, min(j, d), s+=A[i, j-k+1]); A[i+1, j+1]=s)); for(i=d, N, v[i]+=A[i, i-d+1])); for(i=1, N, print1(v[i]", ")) \\ Robert Gerbicz, Apr 06 2011
CROSSREFS
KEYWORD
nonn
AUTHOR
Jack W Grahl, Feb 02 2009
EXTENSIONS
More terms from Alois P. Heinz, Jun 14 2009
Edited by N. J. A. Sloane, Apr 06 2011
STATUS
approved