A229724 - OEIS

%I #18 Sep 28 2013 16:00:12

%S 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 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 38,22,11,5,2,1,45,61,48,29,15,7,3,1,67,82,66,41

%N 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 Row sums are A086543.

%H Alois P. Heinz, <a href="/A229724/b229724.txt">Rows n = 1..200, flattened</a>

%F O.g.f. for column k: x^(2k-1)/[ prod_{j=1..2k-1}(1-x^j)*prod_{j>=k} (1-x^(2j)) ].

%F For even n=2j and k>=ceiling((n+2)/4) T(n,k)=A058695(j-k).

%F For odd n=2j-1 and k>=ceiling((n+2)/4) T(n,k)= A058696(j-k).

%e 1;

%e 1;

%e 2,   1;

%e 2,   1;

%e 4,   2,  1;

%e 4,   3,  1;

%e 7,   5,  2,  1;

%e 7,   6,  3,  1;

%e 12, 10,  5,  2, 1;

%e 12, 12,  7,  3, 1;

%e 19, 18, 11,  5, 2, 1;

%e 19, 22, 14,  7, 3, 1;

%e 30, 31, 21, 11, 5, 2, 1;

%e 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 b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i=1, 1+x,

%p        b(n, i-1) +`if`(i>n, 0, (p->`if`(irem(i, 2, 'r')=0, p,

%p        coeff(p, x, 0)*(1+x^(r+1)) +add(coeff(p, x, j)*x^j,

%p        j=r+2..degree(p))))(b(n-i, i)))))

%p     end:

%p T:= n->(p-> seq(coeff(p, x, j), j=1..degree(p)))(b(n, n)):

%p seq(T(n), n=1..20);  # _Alois P. Heinz_, Sep 28 2013

%t 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 Column k=1 gives: A025065(n-1) for n>1.

%K nonn,tabf

%O 1,3

%A _Geoffrey Critzer_, Sep 28 2013