A118198 Triangle read by rows: T(n,k) is the number of partitions of n having k parts equal to the size of the Durfee square (0<=k<=n).

1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 2, 0, 1, 1, 2, 2, 1, 0, 1, 2, 3, 2, 2, 1, 0, 1, 3, 4, 3, 2, 1, 1, 0, 1, 5, 6, 4, 2, 2, 1, 1, 0, 1, 7, 8, 5, 4, 2, 1, 1, 1, 0, 1, 10, 11, 8, 5, 2, 2, 1, 1, 1, 0, 1, 13, 15, 11, 7, 3, 2, 1, 1, 1, 1, 0, 1, 18, 20, 16, 9, 5, 2, 2, 1, 1, 1, 1, 0, 1, 23, 27, 21, 13, 6, 3, 2, 1

Offset: 0

Author

Emeric Deutsch, Apr 14 2006

Keywords

• nonn
• tabl

Comments

Row sums yield the partition numbers (A000041). T(n,0)=A118199(n)

Examples

			T(4,0)=1 because [4] has Durfee square of size 1 and there is no part equal to 1; T(4,1)=1 because [3,1] has Durfee square of size 1 and there is 1 part equal to 1; T(4,2)=2 because [2,2] has Durfee square of size 2 and there are 2 parts equal to 2 and [2,1,1] has Durfee square of size 1 and there are 2 parts equal to 1; T(4,3)=0 because obviously no partition of 4 can have exactly 3 parts of the same size; T(4,4)=1 because [1,1,1,1] has Durfee square of size 1 and there are 4 parts equal to 1.
Triangle starts:
1;
0,1;
1,0,1;
1,1,0,1;
1,1,2,0,1;
1,2,2,1,0,1;
		

Crossrefs

Cf. A000041, A118199.

Programs

Maple

g:=1+sum(x^(k^2)*sum(t^(k-j)*x^j/product(1-x^i,i=1..j),j=0..k)/(1-t*x^k)/product(1-x^i,i=1..k-1),k=1..20): gser:=simplify(series(g,x=0,30)): P[0]:=1: for n from 1 to 13 do P[n]:=sort(coeff(gser,x^n)) od: for n from 0 to 13 do seq(coeff(P[n],t,p),p=0..n) od; # yields sequence in triangular form

Formula

G.f.=G(t,x)=1+sum(x^(k^2)*P(k-1)sum(t^(k-j)*x^j*P(j), j=0..k)/(1-tx^k), k=1..infinity), where P(m)=1/product(1-x^i,i=1..m).

Extensions

Keyword tabl added by Michel Marcus, Apr 09 2013