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 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. 5
 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 (list; graph; refs; listen; history; text; internal format)
 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 A156041 gives the full array A(n, k). See also A156039, A156040 and A156042. One of two bisections of A188624 (see also A188625). Sequence in context: A320567 A135339 A148170 * A268322 A148171 A318644 Adjacent sequences:  A156040 A156041 A156042 * A156044 A156045 A156046 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

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Last modified January 19 08:18 EST 2022. Contains 350464 sequences. (Running on oeis4.)