%I #25 Nov 25 2015 21:48:29
%S 0,0,1,1,1,1,1,1,1,1,1,2,1,1,2,1,1,1,2,2,1,1,1,2,2,2,1,1,1,2,2,3,2,1,
%T 1,2,2,3,3,2,1,1,2,2,3,4,3,1,1,1,2,2,3,4,4,3,1,1,1,2,2,3,4,5,4,3,1,1,
%U 1,2,2,3,4,5,5,5,3,1,1,2,2,3,4,5,6,6,5,2,1,1,2,2,3,4,5,6,7,7,5,2
%N Irregular triangle T(n,k) read by rows: T(n,k) = number of partitions of n into at least two distinct parts, where the largest part is n-k.
%C Alternate name: T(n,k) = the number of ways that at least two distinct positive integers sum to n, where the largest of these integers is n-k.
%C Row sums = A111133(n).
%C Row sums {k <= floor((n-1)/2)} = A026906(n)
%C Row sums {k > floor((n-1)/2)} = A258259(n)
%F Given T(1,1) = T(2,1) = 0, to find row n>=3: Let k" be the maximum value of k in row g<n, F be floor((n-1)/2) and S(g) be the sum of row g. Then:
%F T(n,k) = S(g)+1 g=k when g<=F (equivalent to A000009(g));
%F T(n,k) = Sum_{j=2*(g-F)-1..k"} T(g,j) g=k when g>F, 2*(g-F)-1 <= k" and n is even;
%F T(n,k) = Sum_{j=2*(g-F)..k"} T(g,j) g=k when g>F, 2*(g-F) <= k" and n is odd.
%e Triangle starts T(1,1):
%e n/k 1 2 3 4 5 6 7 8 9 10 11 12 13 14
%e 1 0
%e 2 0
%e 3 1
%e 4 1
%e 5 1 1
%e 6 1 1 1
%e 7 1 1 2
%e 8 1 1 2 1
%e 9 1 1 2 2 1
%e 10 1 1 2 2 2 1
%e 11 1 1 2 2 3 2
%e 12 1 1 2 2 3 3 2
%e 13 1 1 2 2 3 4 3 1
%e 14 1 1 2 2 3 4 4 3 1
%e 15 1 1 2 2 3 4 5 4 3 1
%e 16 1 1 2 2 3 4 5 5 5 3
%e 17 1 1 2 2 3 4 5 6 6 5 2
%e 18 1 1 2 2 3 4 5 6 7 7 5 2
%e 19 1 1 2 2 3 4 5 6 8 8 7 5 1
%e 20 1 1 2 2 3 4 5 6 8 9 9 8 4 1
%e T(15,8) = 4: the four partitions of 15 into at least two distinct parts with largest part 15-8 = 7 are {7,6,2}; {7,5,3}; {7,5,2,1} and {7,4,3,1}.
%e T(14,k) for k=1..F, with F = floor(13/2) = 6: T(14,1) = 0+1 = 1; T(14,2) = 0+1 = 1; T(14,3) = 1+1 = 2; T(14,4) = 1+1 = 2; T(14,5) = 2+1 = 3; T(14,6) = 3+1 = 4.
%e T(14,k) for k>6: T(14,7) = T(7,1)+T(7,2)+T(7,3) = 1+1+2 = 4; T(14,8) = T(8,3)+T(8,4) = 2+1 = 3; T(14,9) = T(9,5) = 1.
%Y Cf. A000009, A026906, A111133, A258259.
%K nonn,tabf
%O 1,12
%A _Bob Selcoe_, Oct 04 2015