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A097304 Triangle of numbers of partitions of n with m parts which are all odd.

%I A097304 #21 Jan 13 2024 11:34:07
%S A097304 1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,2,0,1,0,1,1,0,2,0,1,0,1,0,2,0,2,0,1,
%T A097304 0,1,1,0,3,0,2,0,1,0,1,0,3,0,3,0,2,0,1,0,1,1,0,4,0,3,0,2,0,1,0,1,0,3,
%U A097304 0,5,0,3,0,2,0,1,0,1,1,0,5,0,5,0,3,0,2,0,1,0,1,0,4,0,6,0,5,0,3,0,2,0,1,0,1
%N A097304 Triangle of numbers of partitions of n with m parts which are all odd.
%H A097304 Álvar Ibeas, <a href="/A097304/b097304.txt">First 100 rows, flattened</a>
%H A097304 W. Lang, <a href="/A097304/a097304.txt">First 10 rows</a>.
%F A097304 T(n, m) := 0 if 1 <= n < m, else T(n, m) = number of partitions of n with m parts which are all odd. Hence T(2*k, 2*j-1) = 0, k >= 1, k >= j >= 1; T(2*k-1, 2*j) = 0, k >= 1, k-1 >= j >= 1.
%F A097304 G.f.: 1/Product_{j>=1} (1 - t*x^(2*j-1)). - _Emeric Deutsch_, Feb 24 2006
%F A097304 T(n, k) = T(n-1, k-1) + T(n-2*k, k). If n+k is even, T(n, k) = A008284((n+k)/2, k) = A072233((n-k)/2, k); 0 otherwise. - _Álvar Ibeas_, Jul 25 2020
%e A097304 [1];
%e A097304 [0,1];
%e A097304 [1,0,1];
%e A097304 [0,1,0,1];
%e A097304 [1,0,1,0,1];
%e A097304 [0,2,0,1,0,1];
%e A097304 ...
%e A097304 T(6,2)=2 because 6 = 1+5 = 3+3; T(6,1) = 0 = T(6,3): there are no partitions of 6 into either one or three parts with only odd numbers;
%e A097304 T(6,4)=1 from 6 = 1+1+1+3; T(6,6)=1 from 6 = 1+1+1+1+1+1.
%p A097304 g:=1/product(1-t*x^(2*j-1),j=1..30)-1: gser:=simplify(series(g,x=0,17)): for n from 1 to 15 do P[n]:=sort(coeff(gser,x^n)) od: seq(seq(coeff(P[n],t^j),j=1..n),n=1..15); # _Emeric Deutsch_, Feb 24 2006
%Y A097304 Row sums: A000009 (number of partitions of n into odd parts).
%Y A097304 Cf. A008284 (partitions of n into k parts).
%Y A097304 Cf. A152140.
%K A097304 nonn,tabl,easy
%O A097304 1,17
%A A097304 _Wolfdieter Lang_, Aug 13 2004