A207194 - OEIS

%I #14 Nov 07 2014 05:35:39

%S 1,1,1,1,1,2,2,1,1,1,2,4,6,8,8,6,3,1,1,1,2,4,8,14,23,34,44,50,50,43,

%T 32,20,10,4,1,1,1,2,4,8,16,30,54,91,143,208,280,350,406,436,434,400,

%U 340,265,189,122,70,35,15,5,1,1,1,2,4,8,16,32,62,117,211

%N Triangular array read by rows. T(n,k) is the number of compositions of the integer k into at most n summands, each of which is at most n, n >= 0, k >= 0.

%C Row lengths = n^2 + 1. T(n,0)= 1, the composition of 0 into an empty sequence of summands. T(n,n^2) = 1, the composition of n^2 into exactly n parts all equal to n.

%H Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, Cambridge Univ. Press, 2009, page 45.

%F E.g.f. for row n: B(A(z)) where A(z)= (z-z^(n+1))/(1-z) and B(z)= (1-z^(n+1))/(1-z).

%e Triangle:

%e   1

%e   1    1

%e   1    1    2    2    1

%e   1    1    2    4    6    8    8    6    3   1

%e T(3,5)=8 because there are 8 compositions of 5 into at most 3 parts that are less than or equal to 3: 1+1+3, 1+2+2, 1+3+1, 2+1+2, 2+2+1, 2+3, 3+1+1, 3+2.

%t nn=200; a=(z-z^k)/(1-z); Table[CoefficientList[Series[(1-a^k)/(1-a),{z,0,nn}], z], {k,1,7}]//Flatten

%K nonn,tabf

%O 0,6

%A _Geoffrey Critzer_, Feb 15 2012