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A274199 Limiting reverse row of the array A274190.

%I A274199 #14 Mar 07 2023 02:37:19
%S A274199 1,1,2,3,5,8,12,19,29,44,67,101,152,228,342,511,763,1138,1695,2523,
%T A274199 3752,5578,8287,12307,18272,27119,40241,59700,88556,131340,194772,
%U A274199 288815,428229,634900,941263,1395397,2068560,3066372,4545387,6737633,9987026,14803303
%N A274199 Limiting reverse row of the array A274190.
%C A274199 The triangular array (g(n,k)) at A274190 is defined as follows:  g(n,k) = 1 for n >= 0; g(n,k) = 0 if k > n; g(n,k) = g(n-1,k-1) + g(n-1,2k) for n > 0, k > 1.
%C A274199 From _Gus Wiseman_, Mar 12 2021: (Start)
%C A274199 Also (apparently) the number of compositions of n where all adjacent parts (x, y), satisfy x < 2y. For example, the a(1) = 1 through a(6) = 12 compositions are:
%C A274199   (1)  (2)   (3)    (4)     (5)      (6)
%C A274199        (11)  (12)   (13)    (14)     (15)
%C A274199              (111)  (22)    (23)     (24)
%C A274199                     (112)   (32)     (33)
%C A274199                     (1111)  (113)    (114)
%C A274199                             (122)    (123)
%C A274199                             (1112)   (132)
%C A274199                             (11111)  (222)
%C A274199                                      (1113)
%C A274199                                      (1122)
%C A274199                                      (11112)
%C A274199                                      (111111)
%C A274199 (End)
%H A274199 Daniel Gabric and Jeffrey Shallit, <a href="https://arxiv.org/abs/2302.13147">Smallest and Largest Block Palindrome Factorizations</a>, arXiv:2302.13147 [math.CO], 2023.
%e A274199 Row (g(14,k)):  1, 51, 73, 69, 55, 40, 28, 19, 12, 8, 5, 3, 2, 1, 1; the reversal is 1 1 2 3 5 8 12 19 28 ..., which agrees with A274199 up to 19.
%t A274199 g[n_, 0] = g[n, 0] = 1;
%t A274199 g[n_, k_] := g[n, k] = If[k > n, 0, g[n - 1, k - 1] + g[n - 1, 2 k]];
%t A274199 z = 300; u = Reverse[Table[g[z, k], {k, 0, z}]];
%t A274199 z = 301; v = Reverse[Table[g[z, k], {k, 0, z}]];
%t A274199 w = Join[{1}, Intersection[u, v]] (* A274199 *)
%t A274199 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Table[#[[i]]<2*#[[i-1]],{i,2,Length[#]}]&]],{n,15}] (* _Gus Wiseman_, Mar 12 2021 *)
%Y A274199 Cf. A274190, A274200, A274201.
%Y A274199 Cf. A000929, A003242, A154402, A224957, A342094, A342095, A342096, A342097, A342098, A342191, A342330-A342342.
%K A274199 nonn,easy
%O A274199 0,3
%A A274199 _Clark Kimberling_, Jun 13 2016