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%I #15 Aug 29 2019 16:15:06
%S 1,1,1,2,1,2,1,2,3,1,3,4,1,3,4,5,1,3,5,6,1,4,6,7,8,1,4,7,9,10,1,4,8,
%T 10,11,12,1,5,9,12,14,15,1,5,10,14,16,17,18,1,5,11,16,19,21,22,1,6,13,
%U 19,23,25,26,27,1,6,14,21,26,29,31,32,1,6,15,24,30,34,36,37,38,1,7,17,27
%N Array of number of partitions of n with odd parts not exceeding 2*m-1 with m in {1, 2, ..., ceiling(n/2)}.
%C The sequence of row lengths of this array is A008619 = [1,1,2,2,3,3,4,4,5,5,6,6,7,7,...].
%C This is the partial row sums array of array A097305.
%C The number of partitions of N=2*n (n >= 1) into even parts not exceeding 2*k, with k from {1,...,n}, is given by the triangle A026820(n,k).
%H Wolfdieter Lang, <a href="/A097306/a097306_1.txt">First 18 rows.</a>
%F T(n, m) = number of partitions of n with odd parts only and largest parts <= 2*m-1 for m from {1, 2, ..., ceiling(n/2)}.
%F T(n, m) = Sum_{k=1..m} A097305(n, k), m = 1..ceiling(n/2), n >= 1.
%e [1]; [1]; [1,2]; [1,2]; [1,2,3]; [1,3,4]; [1,3,4,5]; [1,3,5,6]; ...
%e T(8,2)=3 because there are three partitions of 8 with odd parts not exceeding 3, namely (1^8), (1^5,3) and (1^2,3^2).
%e T(6,2)=3 from the partitions (1^6), (1^3,3) and (3^2).
%p Sequence of row numbers for n>=1: [seq(coeff(series(product(1/(1-x^(2*k-1)),k=1..p),x,n+1),x,n),p=1..ceil(n/2))].
%Y Row sums: A097307.
%K nonn,tabf,easy
%O 1,4
%A _Wolfdieter Lang_, Aug 13 2004
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