VOOZH about

URL: https://oeis.org/A059619

⇱ A059619 - OEIS

login
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.
2
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
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: A348028 A382964 A375011 * A098950 A318873 A346403
KEYWORD
nonn,tabl
AUTHOR
Henry Bottomley, Jan 31 2001
STATUS
approved