%I #14 Mar 07 2023 02:37:19
%S 1,1,2,3,5,8,12,19,29,44,67,101,152,228,342,511,763,1138,1695,2523,
%T 3752,5578,8287,12307,18272,27119,40241,59700,88556,131340,194772,
%U 288815,428229,634900,941263,1395397,2068560,3066372,4545387,6737633,9987026,14803303
%N Limiting reverse row of the array A274190.
%C 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 From _Gus Wiseman_, Mar 12 2021: (Start)
%C 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 (1) (2) (3) (4) (5) (6)
%C (11) (12) (13) (14) (15)
%C (111) (22) (23) (24)
%C (112) (32) (33)
%C (1111) (113) (114)
%C (122) (123)
%C (1112) (132)
%C (11111) (222)
%C (1113)
%C (1122)
%C (11112)
%C (111111)
%C (End)
%H 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 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 g[n_, 0] = g[n, 0] = 1;
%t g[n_, k_] := g[n, k] = If[k > n, 0, g[n - 1, k - 1] + g[n - 1, 2 k]];
%t z = 300; u = Reverse[Table[g[z, k], {k, 0, z}]];
%t z = 301; v = Reverse[Table[g[z, k], {k, 0, z}]];
%t w = Join[{1}, Intersection[u, v]] (* A274199 *)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Table[#[[i]]<2*#[[i-1]],{i,2,Length[#]}]&]],{n,15}] (* _Gus Wiseman_, Mar 12 2021 *)
%Y Cf. A274190, A274200, A274201.
%Y Cf. A000929, A003242, A154402, A224957, A342094, A342095, A342096, A342097, A342098, A342191, A342330-A342342.
%K nonn,easy
%O 0,3
%A _Clark Kimberling_, Jun 13 2016