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%I #18 Jan 13 2024 11:33:30
%S 1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0,2,0,1,0,1,0,1,0,2,0,1,
%T 0,1,0,0,2,0,2,0,1,0,1,0,1,0,3,0,2,0,1,0,1,0,0,3,0,3,0,2,0,1,0,1,0,1,
%U 0,4,0,3,0,2,0,1,0,1,0,0,3,0,5,0,3,0,2,0,1,0,1,0,1,0,5,0,5,0,3,0,2,0,1,0,1
%N Triangle read by rows: T(n,k) (n>=0, 0<=k<=n) = number of partitions of n into k odd parts.
%C The number of partitions of n into k odd parts is equal to the number of partitions of (n+k)/2 into k parts; or equivalently the number of partitions of (n-k)/2 into at most k parts. - _Franklin T. Adams-Watters_, Sep 25 2009
%H Alois P. Heinz, <a href="/A152140/b152140.txt">Rows n = 0..140, flattened</a>
%e n= 0, k= 0: [];
%e n= 1, k= 1: [1] ;
%e n= 2, k= 2: [1, 1] ;
%e n= 3, k= 1: [3] ;
%e n= 3, k= 3: [1, 1, 1] ;
%e n= 4, k= 2: [1, 3] ;
%e n= 4, k= 4: [1, 1, 1, 1];
%e n= 5, k= 1: [5];
%e n= 5, k= 3: [1, 1, 3];
%e n= 5, k= 5: [1, 1, 1, 1, 1];
%e n= 6, k= 2: [3, 3] or [1, 5];
%e n= 6, k= 4: [1, 1, 1, 3];
%e n= 6, k= 6: [1, 1, 1, 1, 1, 1];
%e Triangle begins:
%e 1
%e 0 1
%e 0 0 1
%e 0 1 0 1
%e 0 0 1 0 1
%e 0 1 0 1 0 1
%e 0 0 2 0 1 0 1
%e 0 1 0 2 0 1 0 1
%e 0 0 2 0 2 0 1 0 1
%e 0 1 0 3 0 2 0 1 0 1
%e 0 0 3 0 3 0 2 0 1 0 1
%e 0 1 0 4 0 3 0 2 0 1 0 1
%e 0 0 3 0 5 0 3 0 2 0 1 0 1
%e 0 1 0 5 0 5 0 3 0 2 0 1 0 1
%e 0 0 4 0 6 0 5 0 3 0 2 0 1 0 1
%e 0 1 0 7 0 7 0 5 0 3 0 2 0 1 0 1
%e 0 0 4 0 9 0 7 0 5 0 3 0 2 0 1 0 1
%e 0 1 0 8 0 10 0 7 0 5 0 3 0 2 0 1 0 1
%e 0 0 5 0 11 0 11 0 7 0 5 0 3 0 2 0 1 0 1
%e 0 1 0 10 0 13 0 11 0 7 0 5 0 3 0 2 0 1 0 1
%e 0 0 5 0 15 0 14 0 11 0 7 0 5 0 3 0 2 0 1 0 1
%e 0 1 0 12 0 18 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
%e 0 0 6 0 18 0 20 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
%e 0 1 0 14 0 23 0 21 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
%e 0 0 6 0 23 0 26 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
%e 0 1 0 16 0 30 0 28 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
%e 0 0 7 0 27 0 35 0 29 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
%e 0 1 0 19 0 37 0 38 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
%e 0 0 7 0 34 0 44 0 40 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
%e 0 1 0 21 0 47 0 49 0 41 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
%e 0 0 8 0 39 0 58 0 52 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
%e 0 1 0 24 0 57 0 65 0 54 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
%e 0 0 8 0 47 0 71 0 70 0 55 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
%e 0 1 0 27 0 70 0 82 0 73 0 56 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
%e 0 0 9 0 54 0 90 0 89 0 75 0 56 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
%e 0 1 0 30 0 84 0 105 0 94 0 76 0 56 0 42 0 30 0 22 0 15 0 11 0 7 0 5 0 3 0 2 0 1 0 1
%p b:= proc(n, i) option remember; local j; if n=0 then 1
%p elif i<1 then 0 elif irem(i, 2)=0 then b(n, i-1)
%p else []; for j from 0 to n/i do zip((x, y)->x+y, %,
%p [0$j, b(n-i*j, i-2)], 0) od; %[] fi
%p end:
%p T:= n-> b(n$2):
%p seq(T(n), n=0..13); # _Alois P. Heinz_, May 31 2013
%t nn = 10; CoefficientList[
%t Series[Product[1/(1 - y x^i), {i, 1, nn, 2}], {x, 0, nn}], {x, y}] (* _Geoffrey Critzer_, May 31 2013 *)
%Y Cf. A000009 (row sums), A097304, A107379, A152146, A152157.
%K nonn,tabl
%O 0,24
%A _R. J. Mathar_, Sep 25 2009, offset corrected Jul 09 2012
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