A156043 A(n,n), where A(n,k) is the number of compositions (ordered partitions) of n into k parts (parts of size 0 being allowed), with the first part being greater than or equal to all the rest.
1, 2, 4, 11, 32, 102, 331, 1101, 3724, 12782, 44444, 156334, 555531, 1991784, 7197369, 26186491, 95847772, 352670170, 1303661995, 4838822931, 18025920971, 67371021603, 252538273442, 949164364575, 3576145084531, 13503991775252
Offset: 1
Keywords
Examples
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)
Links
- Robert Gerbicz, Table of n, a(n) for n = 1..500
Crossrefs
Programs
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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
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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 *) -
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
Extensions
More terms from Alois P. Heinz, Jun 14 2009
Edited by N. J. A. Sloane, Apr 06 2011
Comments