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%I #14 Sep 11 2012 07:21:20
%S 1,1,1,1,0,1,3,1,1,1,4,2,0,1,1,6,2,1,1,1,1,10,4,2,1,1,1,1,15,6,3,1,2,
%T 1,1,1,21,9,4,2,1,2,1,1,1,30,12,6,3,2,2,2,1,1,1,43,18,8,5,3,2,2,2,1,1,
%U 1,59,25,12,6,3,3,3,2,2,1,1,1,82,34,17,9,5,4,3,3,2,2,1,1,1,111,48,22,12
%N As upper right triangle, number of strongly unimodal partitions of n (strongly unimodal means strictly increasing then strictly decreasing) where initial part is k.
%H R. C. Rhoades, <a href="http://math.stanford.edu/~rhoades/FILES/unimodal.pdf">Strongly Unimodal Sequences and Mixed Mock Modular Forms</a>
%F T(n, k)=S(n, k)+sum_j[T(n-k, j)] for j>k, where S(n, k)=A059607(n, k)=sum_j[S(n-k, j)] for k>j [note reversal] with S(0, 0)=1.
%e Rows start:
%e 1, 1, 1, 3, 4, 6, 10, 15, 21, 30, 43, 59, 82, 111, ...
%e 1, 0, 1, 2, 2, 4, 6, 9, 12, 18, 25, 34, 48, ...
%e 1, 1, 0, 1, 2, 3, 4, 6, 8, 12, 17, 22, ...
%e 1, 1, 1, 1, 1, 2, 3, 5, 6, 9, 12, ...
%e 1, 1, 1, 2, 1, 2, 3, 3, 5, ...
%e 1, 1, 1, 2, 2, 2, 3, 4, ...
%e 1, 1, 1, 2, 2, 3, 3, ...
%e 1, 1, 1, 2, 2, 3, ...
%e 1, 1, 1, 2, 2, ...
%e 1, 1, 1, 2, ...
%e 1, 1, 1, ...
%e 1, 1, ...
%e 1, ... etc.
%e T(16,6)=8 since 16 can be written as 6+10, 6+9+1, 6+8+2, 6+7+3, 6+7+2+1, 6+5+4+1, 6+5+3+2, or 6+4+3+2+1 (but for example neither 6+6+4 nor 6+8+1+1 which are only weakly unimodal).
%t s[n_?Positive, k_] := s[n, k] = Sum[s[n-k, j], {j, 0, k-1}]; s[0, 0] = 1; s[0, _] = 0; s[_?Negative, _] = 0; t[n_, k_] := t[n, k] = s[n, k] + Sum[t[n-k, j], {j, k+1, n}]; Table[t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 11 2012 *)
%Y Top row is A059618 and is sum of other rows (for n>0). Cf. A000009, A000041, A001523, A059607.
%K nonn,tabl
%O 0,7
%A _Henry Bottomley_, Jan 31 2001
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