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A115604 Triangle read by rows: T(n,k) is the number of partitions of n into odd parts in which the smallest part occurs k times (1<=k<=n).

%I A115604 #3 Mar 30 2012 17:36:07
%S A115604 1,0,1,1,0,1,1,0,0,1,1,1,0,0,1,1,1,1,0,0,1,2,1,0,1,0,0,1,2,1,1,0,1,0,
%T A115604 0,1,2,1,2,1,0,1,0,0,1,3,2,1,1,1,0,1,0,0,1,3,3,1,1,1,1,0,1,0,0,1,4,2,
%U A115604 2,2,1,1,1,0,1,0,0,1,5,3,2,2,1,1,1,1,0,1,0,0,1,5,4,3,2,2,1,1,1,1,0,1,0,0,1
%N A115604 Triangle read by rows: T(n,k) is the number of partitions of n into odd parts in which the smallest part occurs k times (1<=k<=n).
%C A115604 Row sums yield A000009. T(n,1)=A087897(n+2). Sum(k*T(n,k),k=1..n)=A092268(n).
%F A115604 G.f.=G(t,x)=sum(tx^(2k-1)/[(1-tx^(2k-1))product(1-x^(2i-1), i=k+1..infinity)], k=1..infinity).
%e A115604 T(14,2)=4 because we have [9,3,1,1],[7,7],[7,5,1,1] and [3,3,3,3,1,1].
%e A115604 Triangle starts:
%e A115604 1;
%e A115604 0,1;
%e A115604 1,0,1;
%e A115604 1,0,0,1;
%e A115604 1,1,0,0,1;
%e A115604 1,1,1,0,0,1;
%e A115604 2,1,0,1,0,0,1;
%p A115604 g:=sum(t*x^(2*k-1)/(1-t*x^(2*k-1))/product(1-x^(2*i-1),i=k+1..40),k=1..40): gser:=simplify(series(g,x=0,55)): for n from 1 to 15 do P[n]:=expand(coeff(gser,x^n)) od: for n from 1 to 15 do seq(coeff(P[n],t^j),j=1..n) od; # yields sequence in triangular form
%Y A115604 Cf. A000009, A087897, A092268, A117408.
%K A115604 nonn,tabl
%O A115604 1,22
%A A115604 _Emeric Deutsch_, Mar 13 2006