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A262885 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.

%I A262885 #25 Nov 25 2015 21:48:29
%S A262885 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 A262885 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 A262885 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 A262885 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 A262885 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 A262885 Row sums = A111133(n).
%C A262885 Row sums {k <= floor((n-1)/2)} = A026906(n)
%C A262885 Row sums {k > floor((n-1)/2)} = A258259(n)
%F A262885 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 A262885 T(n,k) = S(g)+1 g=k when g<=F (equivalent to A000009(g));
%F A262885 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 A262885 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 A262885 Triangle starts T(1,1):
%e A262885 n/k  1 2 3 4 5 6 7 8 9 10 11 12 13 14
%e A262885 1    0
%e A262885 2    0
%e A262885 3    1
%e A262885 4    1
%e A262885 5    1 1
%e A262885 6    1 1 1
%e A262885 7    1 1 2
%e A262885 8    1 1 2 1
%e A262885 9    1 1 2 2 1
%e A262885 10   1 1 2 2 2 1
%e A262885 11   1 1 2 2 3 2
%e A262885 12   1 1 2 2 3 3 2
%e A262885 13   1 1 2 2 3 4 3 1
%e A262885 14   1 1 2 2 3 4 4 3 1
%e A262885 15   1 1 2 2 3 4 5 4 3 1
%e A262885 16   1 1 2 2 3 4 5 5 5 3
%e A262885 17   1 1 2 2 3 4 5 6 6 5  2
%e A262885 18   1 1 2 2 3 4 5 6 7 7  5  2
%e A262885 19   1 1 2 2 3 4 5 6 8 8  7  5  1
%e A262885 20   1 1 2 2 3 4 5 6 8 9  9  8  4  1
%e A262885 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 A262885 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 A262885 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 A262885 Cf. A000009, A026906, A111133, A258259.
%K A262885 nonn,tabf
%O A262885 1,12
%A A262885 _Bob Selcoe_, Oct 04 2015