A059619 As upper right triangle, number of strongly unimodal partitions of n (strongly unimodal means strictly increasing then strictly decreasing) where initial part is k.
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, 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, 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
Offset: 0
Examples
Rows start: 1, 1, 1, 3, 4, 6, 10, 15, 21, 30, 43, 59, 82, 111, ... 1, 0, 1, 2, 2, 4, 6, 9, 12, 18, 25, 34, 48, ... 1, 1, 0, 1, 2, 3, 4, 6, 8, 12, 17, 22, ... 1, 1, 1, 1, 1, 2, 3, 5, 6, 9, 12, ... 1, 1, 1, 2, 1, 2, 3, 3, 5, ... 1, 1, 1, 2, 2, 2, 3, 4, ... 1, 1, 1, 2, 2, 3, 3, ... 1, 1, 1, 2, 2, 3, ... 1, 1, 1, 2, 2, ... 1, 1, 1, 2, ... 1, 1, 1, ... 1, 1, ... 1, ... etc. 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).
Links
- R. C. Rhoades, Strongly Unimodal Sequences and Mixed Mock Modular Forms
Crossrefs
Programs
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Mathematica
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 *)
Formula
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.