%N Triangle read by rows: T(n,k) is the number of partitions of n into odd parts such that the smallest part is k (n>=1, k>=1).
%C Row 2n-1 has 2n-1 terms; row 4n+2 has 2n+1 terms; row 4n has 2n-1 terms. Row sums yield A000009. T(n,2k)=0. Sum(k*T(n,k),k>=1)=A092314(n)
%F G.f.=sum(t^(2j-1)*x^(2j-1)/product(1-x^(2i-1), i=j..infinity), j=1..infinity).
%e T(12,3)=2 because we have [9,3] and [3,3,3,3].
%e Triangle starts:
%p g:=sum(t^(2*j-1)*x^(2*j-1)/product(1-x^(2*i-1),i=j..20),j=1..30): gser:=simplify(series(g,x=0,20)): for n from 1 to 17 do P[n]:=sort(coeff(gser,x^n)) od: d:=proc(n) if n mod 2 = 1 then n elif n mod 4 = 2 then n/2 else n/2-1 fi end: for n from 1 to 17 do seq(coeff(P[n],t^j),j=1..d(n)) od; # yields sequence in triangular form; d(n) is the degree of the polynomial P[n]
%Y Cf. A000009, A092314, A116799.
%A _Emeric Deutsch_, Feb 24 2006