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A229724 Triangular array read by rows: T(n,k) is the number of partitions of n in which the greatest odd part is equal to 2k-1; n >= 1, 1 <= k <= ceiling(n/2).

%I A229724 #18 Sep 28 2013 16:00:12
%S A229724 1,1,2,1,2,1,4,2,1,4,3,1,7,5,2,1,7,6,3,1,12,10,5,2,1,12,12,7,3,1,19,
%T A229724 18,11,5,2,1,19,22,14,7,3,1,30,31,21,11,5,2,1,30,37,27,15,7,3,1,45,52,
%U A229724 38,22,11,5,2,1,45,61,48,29,15,7,3,1,67,82,66,41
%N A229724 Triangular array read by rows: T(n,k) is the number of partitions of n in which the greatest odd part is equal to 2k-1; n >= 1, 1 <= k <= ceiling(n/2).
%C A229724 Row sums are A086543.
%H A229724 Alois P. Heinz, <a href="/A229724/b229724.txt">Rows n = 1..200, flattened</a>
%F A229724 O.g.f. for column k: x^(2k-1)/[ prod_{j=1..2k-1}(1-x^j)*prod_{j>=k} (1-x^(2j)) ].
%F A229724 For even n=2j and k>=ceiling((n+2)/4) T(n,k)=A058695(j-k).
%F A229724 For odd n=2j-1 and k>=ceiling((n+2)/4) T(n,k)= A058696(j-k).
%e A229724 1;
%e A229724 1;
%e A229724 2,   1;
%e A229724 2,   1;
%e A229724 4,   2,  1;
%e A229724 4,   3,  1;
%e A229724 7,   5,  2,  1;
%e A229724 7,   6,  3,  1;
%e A229724 12, 10,  5,  2, 1;
%e A229724 12, 12,  7,  3, 1;
%e A229724 19, 18, 11,  5, 2, 1;
%e A229724 19, 22, 14,  7, 3, 1;
%e A229724 30, 31, 21, 11, 5, 2, 1;
%e A229724 T(7,2) = 5 because we have: 4+3 = 3+3+1 = 3+2+2 = 3+2+1+1 = 3+1+1+1+1.
%p A229724 b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i=1, 1+x,
%p A229724        b(n, i-1) +`if`(i>n, 0, (p->`if`(irem(i, 2, 'r')=0, p,
%p A229724        coeff(p, x, 0)*(1+x^(r+1)) +add(coeff(p, x, j)*x^j,
%p A229724        j=r+2..degree(p))))(b(n-i, i)))))
%p A229724     end:
%p A229724 T:= n->(p-> seq(coeff(p, x, j), j=1..degree(p)))(b(n, n)):
%p A229724 seq(T(n), n=1..20);  # _Alois P. Heinz_, Sep 28 2013
%t A229724 nn=16;Map[Select[#,#>0&]&,Drop[Transpose[Table[CoefficientList[Series[x^(2k-1)/Product[1-x^j,{j,1,2k-1}] /Product[(1-x^(2j)),{j,k,nn}],{x,0,nn}],x],{k,1,nn/2}]],1]]//Grid
%Y A229724 Column k=1 gives: A025065(n-1) for n>1.
%K A229724 nonn,tabf
%O A229724 1,3
%A A229724 _Geoffrey Critzer_, Sep 28 2013