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%I #13 Dec 19 2015 10:57:56
%S 1,1,1,2,1,1,4,2,1,1,8,3,2,1,1,15,5,3,2,1,1,27,8,5,3,2,1,1,47,13,7,5,
%T 3,2,1,1,79,21,11,7,5,3,2,1,1,130,33,16,11,7,5,3,2,1,1,209,52,24,15,
%U 11,7,5,3,2,1,1,330,80,35,22,15,11,7,5,3,2,1,1,512,122,52,31,22,15,11,7,5,3
%N As upper right triangle, number of weakly unimodal partitions of n where initial part is k (n >= k >= 1).
%C Weakly unimodal means nondecreasing then nonincreasing.
%F T(n, k) = S(n, k) - S(n-k, k) + Sum_j[T(n-k, j)] for j >= k, where S(n, k) = A008284(n, k) = Sum_j[S(n-k, j)] for n>k >= j [note reversal] with S[n, n] = 1.
%e Rows are {1,1,2,4,8,15,...}, {1,1,2,3,5,8,...}, {1,1,2,3,5,7,...} etc.
%e As an upper right triangle:
%e 1, 1, 2, 4, 8, 15, ...,
%e 1, 1, 2, 3, 5, 8, ...,
%e 1, 1, 2, 3, 5, 7, ...,
%e ...
%e As a left downward triangle, it starts:
%e 1;
%e 1, 1;
%e 2, 1, 1;
%e 4, 2, 1, 1;
%e 8, 3, 2, 1, 1;
%e 15, 5, 3, 2, 1, 1;
%e 27, 8, 5, 3, 2, 1, 1;
%e ...
%e T(9,3)=11 since 9 can be written as 3+6, 3+5+1, 3+4+2, 3+4+1+1, 3+3+3, 3+3+2+1, 3+3+1+1+1, 3+2+2+2, 3+2+2+1+1, 3+2+1+1+1+1 or 3+1+1+1+1+1.
%Y Column sums give A001523. Cf. A008284, A026836, A008284, A059607, A059619.
%K nonn,tabl
%O 1,4
%A _Henry Bottomley_, Feb 01 2001
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