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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).

%I A118198 #7 Apr 09 2013 08:08:26
%S A118198 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,
%T A118198 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,
%U A118198 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
%N 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).
%C A118198 Row sums yield the partition numbers (A000041). T(n,0)=A118199(n)
%F A118198 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).
%e A118198 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.
%e A118198 Triangle starts:
%e A118198 1;
%e A118198 0,1;
%e A118198 1,0,1;
%e A118198 1,1,0,1;
%e A118198 1,1,2,0,1;
%e A118198 1,2,2,1,0,1;
%p A118198 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
%Y A118198 Cf. A000041, A118199.
%K A118198 nonn,tabl
%O A118198 0,13
%A A118198 _Emeric Deutsch_, Apr 14 2006
%E A118198 Keyword tabl added by _Michel Marcus_, Apr 09 2013