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A116857 Triangle read by rows: T(n,k) is the number of partitions of n into distinct odd parts, the largest of which is k (n>=1, k>=1).

%I A116857 #9 Jan 19 2022 18:34:11
%S A116857 1,0,0,0,1,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,1,0,1,0,0,
%T A116857 0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,
%U A116857 0,1,0,1,0,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,1,0,0,0,0,0,0,1,0,1,0,1,0,0,0,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,2,0,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,1,0,2,0,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,2,0,1,0,1,0,1,0,1
%N A116857 Triangle read by rows: T(n,k) is the number of partitions of n into distinct odd parts, the largest of which is k (n>=1, k>=1).
%C A116857 Both rows 2n-1 and 2n have 2n-1 terms each. Row sums yield A000700. T(n,2k)=0 Sum(k*T(n,k),k>=1) = A092316(n).
%H A116857 Alois P. Heinz, <a href="/A116857/b116857.txt">Rows n = 1..350, flattened</a>
%F A116857 G.f.: sum(t^(2j-1)*x^(2j-1)*product(1+x^(2i-1), i=1..j-1), j=1..infinity).
%e A116857 T(20,11)=2 because we have [11,9] and [11,5,3,1].
%e A116857 T(30,17)=3 because we have [17,13],[17,9,3,1] and [17,7,5,1].
%e A116857 Triangle starts:
%e A116857   1;
%e A116857   0;
%e A116857   0,0,1;
%e A116857   0,0,1;
%e A116857   0,0,0,0,1;
%e A116857   0,0,0,0,1;
%e A116857   0,0,0,0,0,0,1;
%e A116857   0,0,0,0,1,0,1;
%e A116857   ...
%p A116857 g:=sum(t^(2*j-1)*x^(2*j-1)*product(1+x^(2*i-1),i=1..j-1),j=1..30): gser:=simplify(series(g,x=0,22)): for n from 1 to 20 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 20 do seq(coeff(P[n],t^j),j=1..2*ceil(n/2)-1) od; # yields sequence in triangular form
%Y A116857 Cf. A000700, A092316.
%K A116857 nonn,tabf
%O A116857 1,137
%A A116857 _Emeric Deutsch_, Feb 24 2006