A207194 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.

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, 32, 20, 10, 4, 1, 1, 1, 2, 4, 8, 16, 30, 54, 91, 143, 208, 280, 350, 406, 436, 434, 400, 340, 265, 189, 122, 70, 35, 15, 5, 1, 1, 1, 2, 4, 8, 16, 32, 62, 117, 211

Offset

0,6

Comments

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.

Links

Table of n, a(n) for n=0..70.

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 45.

Formula

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).

Example

Triangle:
  1
  1    1
  1    1    2    2    1
  1    1    2    4    6    8    8    6    3   1
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.

Mathematica

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

Crossrefs

Sequence in context: A267632 A112465 A112468 * A349670 A086275 A066855

Adjacent sequences: A207191 A207192 A207193 * A207195 A207196 A207197

Keyword

nonn,tabf

Author

Geoffrey Critzer, Feb 15 2012

Status

approved