%I #16 Jun 19 2023 17:08:28

%S 1,2,6,12,28,54,119,230,488,948,1979,3860,7978,15624,32072,63014,

%T 128746,253588,516346,1019072,2069590,4091174,8291746,16412668,

%U 33210428,65808044,132985161,263755984,532421062,1056789662,2131312530,4233176854

%N Number of 1 X (n+1) 0..1 arrays with every row least squares fitting to a positive-slope straight line and every column least squares fitting to a zero- or positive-slope straight line, with a single point array taken as having zero slope.

%C From _Gus Wiseman_, Jun 16 2023: (Start)

%C Also appears to be the number of integer compositions of n + 2 with weighted sum greater than reverse-weighted sum, where the weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i * y_i, and the reverse is Sum_{i=1..k} i * y_{k-i+1}. The a(1) = 1 through a(4) = 12 compositions are:

%C (21) (31) (32) (42)

%C (211) (41) (51)

%C (221) (231)

%C (311) (312)

%C (1211) (321)

%C (2111) (411)

%C (1311)

%C (2121)

%C (2211)

%C (3111)

%C (12111)

%C (21111)

%C The version for partitions is A144300, strict A111133.

%C (End)

%H R. H. Hardin, <a href="/A222970/b222970.txt">Table of n, a(n) for n = 1..210</a>

%e Some solutions for n=3:

%e 0 1 0 1 0 1 1 1 0 0 1 0 0 0 1 1 0 0 0 1

%Y For >= instead of > we have A222855.

%Y The case of equality is A222955.

%Y Row 1 of A222969.

%Y A053632 counts compositions by weighted sum (or reverse-weighted sum).

%Y A264034 counts partitions by weighted sum, reverse A358194.

%Y A304818 gives weighted sum of prime indices, reverse A318283.

%Y Cf. A000005, A000041, A138364, A320387, A360672, A360675, A363626.

%K nonn

%O 1,2

%A _R. H. Hardin_, Mar 10 2013