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A222970 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.

%I A222970 #16 Jun 19 2023 17:08:28
%S A222970 1,2,6,12,28,54,119,230,488,948,1979,3860,7978,15624,32072,63014,
%T A222970 128746,253588,516346,1019072,2069590,4091174,8291746,16412668,
%U A222970 33210428,65808044,132985161,263755984,532421062,1056789662,2131312530,4233176854
%N A222970 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 A222970 From _Gus Wiseman_, Jun 16 2023: (Start)
%C A222970 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 A222970   (21)  (31)   (32)    (42)
%C A222970         (211)  (41)    (51)
%C A222970                (221)   (231)
%C A222970                (311)   (312)
%C A222970                (1211)  (321)
%C A222970                (2111)  (411)
%C A222970                        (1311)
%C A222970                        (2121)
%C A222970                        (2211)
%C A222970                        (3111)
%C A222970                        (12111)
%C A222970                        (21111)
%C A222970 The version for partitions is A144300, strict A111133.
%C A222970 (End)
%H A222970 R. H. Hardin, <a href="/A222970/b222970.txt">Table of n, a(n) for n = 1..210</a>
%e A222970 Some solutions for n=3:
%e A222970   0 1 0 1    0 1 1 1    0 0 1 0    0 0 1 1    0 0 0 1
%Y A222970 For >= instead of > we have A222855.
%Y A222970 The case of equality is A222955.
%Y A222970 Row 1 of A222969.
%Y A222970 A053632 counts compositions by weighted sum (or  reverse-weighted sum).
%Y A222970 A264034 counts partitions by weighted sum, reverse A358194.
%Y A222970 A304818 gives weighted sum of prime indices, reverse A318283.
%Y A222970 Cf. A000005, A000041, A138364, A320387, A360672, A360675, A363626.
%K A222970 nonn
%O A222970 1,2
%A A222970 _R. H. Hardin_, Mar 10 2013