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

0,7

Links

Table of n, a(n) for n=0..94.

R. C. Rhoades, Strongly Unimodal Sequences and Mixed Mock Modular Forms

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.

Example

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).

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 *)

Crossrefs

Top row is A059618 and is sum of other rows (for n>0). Cf. A000009, A000041, A001523, A059607.

Sequence in context: A270825 A364096 A348028 * A098950 A318873 A346403

Adjacent sequences: A059616 A059617 A059618 * A059620 A059621 A059622

Keyword

nonn,tabl

Author

Henry Bottomley, Jan 31 2001

Status

approved