cp's OEIS Frontend

A059623 As upper right triangle, number of weakly unimodal partitions of n where initial part is k (n >= k >= 1).

%I A059623 #13 Dec 19 2015 10:57:56
%S A059623 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 A059623 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 A059623 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 A059623 As upper right triangle, number of weakly unimodal partitions of n where initial part is k (n >= k >= 1).
%C A059623 Weakly unimodal means nondecreasing then nonincreasing.
%F A059623 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 A059623 Rows are {1,1,2,4,8,15,...}, {1,1,2,3,5,8,...}, {1,1,2,3,5,7,...} etc.
%e A059623 As an upper right triangle:
%e A059623   1,  1,  2,  4,  8, 15, ...,
%e A059623       1,  1,  2,  3,  5,  8, ...,
%e A059623           1,  1,  2,  3,  5,  7, ...,
%e A059623               ...
%e A059623 As a left downward triangle, it starts:
%e A059623    1;
%e A059623    1, 1;
%e A059623    2, 1, 1;
%e A059623    4, 2, 1, 1;
%e A059623    8, 3, 2, 1, 1;
%e A059623   15, 5, 3, 2, 1, 1;
%e A059623   27, 8, 5, 3, 2, 1, 1;
%e A059623   ...
%e A059623 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 A059623 Column sums give A001523. Cf. A008284, A026836, A008284, A059607, A059619.
%K A059623 nonn,tabl
%O A059623 1,4
%A A059623 _Henry Bottomley_, Feb 01 2001