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A144406 Rectangular array A read by upward antidiagonals: entry A(n,k) in row n and column k gives the number of compositions of k in which no part exceeds n, n>=1, k>=0.

%I A144406 #47 Jun 29 2025 10:58:26
%S A144406 1,1,1,1,1,1,1,1,2,1,1,1,2,3,1,1,1,2,4,5,1,1,1,2,4,7,8,1,1,1,2,4,8,13,
%T A144406 13,1,1,1,2,4,8,15,24,21,1,1,1,2,4,8,16,29,44,34,1,1,1,2,4,8,16,31,56,
%U A144406 81,55,1,1,1,2,4,8,16,32,61,108,149,89,1,1,1,2,4,8,16,32,63,120,208,274,144,1
%N A144406 Rectangular array A read by upward antidiagonals: entry A(n,k) in row n and column k gives the number of compositions of k in which no part exceeds n, n>=1, k>=0.
%C A144406 Polynomial expansion as antidiagonal of p(x,n) = (x-1)/(x^n*(-x+(2*x-1)/x^n)). Based on the Pisot general polynomial type q(x,n) = x^n - (x^n-1)/(x-1) (the original name of the sequence).
%C A144406 Row sums are 1, 2, 3, 5, 8, 14, ... (A079500).
%C A144406 Conjecture: Since the array row sequences successively tend to A000079, the absolute values of nonzero differences between two successive row sequences tend to A045623 = {1,2,5,12,28,64,144,320,704,1536,...}, as k -> infinity. - _L. Edson Jeffery_, Dec 26 2013
%F A144406 t(n,m) = antidiagonal_expansion of p(x,n) where p(x,n) = (x-1)/(x^n*(-x+(2*x-1)/x^n)).
%F A144406 G.f. for array A: (1-x)/(1 - 2*x + x^(n+1)), n>=1. - _L. Edson Jeffery_, Dec 26 2013
%e A144406 Array A begins:
%e A144406   {1, 1, 1, 1, 1,  1,  1,  1,   1,   1,   1, ...}
%e A144406   {1, 1, 2, 3, 5,  8, 13, 21,  34,  55,  89, ...}
%e A144406   {1, 1, 2, 4, 7, 13, 24, 44,  81, 149, 274, ...}
%e A144406   {1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, ...}
%e A144406   {1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, ...}
%e A144406   {1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, ...}
%e A144406   {1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, ...}
%e A144406   {1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, ...}
%e A144406   {1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, ...}
%e A144406   ... - _L. Edson Jeffery_, Dec 26 2013
%e A144406 As a triangle:
%e A144406   {1},
%e A144406   {1, 1},
%e A144406   {1, 1, 1},
%e A144406   {1, 1, 2, 1},
%e A144406   {1, 1, 2, 3, 1},
%e A144406   {1, 1, 2, 4, 5, 1},
%e A144406   {1, 1, 2, 4, 7, 8, 1},
%e A144406   {1, 1, 2, 4, 8, 13, 13, 1},
%e A144406   {1, 1, 2, 4, 8, 15, 24, 21, 1},
%e A144406   {1, 1, 2, 4, 8, 16, 29, 44, 34, 1},
%e A144406   {1, 1, 2, 4, 8, 16, 31, 56, 81, 55, 1},
%e A144406   {1, 1, 2, 4, 8, 16, 32, 61, 108, 149, 89, 1},
%e A144406   {1, 1, 2, 4, 8, 16, 32, 63, 120, 208, 274, 144, 1},
%e A144406   {1, 1, 2, 4, 8, 16, 32, 64, 125, 236, 401, 504, 233, 1},
%e A144406   {1, 1, 2, 4, 8, 16, 32, 64, 127, 248, 464, 773, 927, 377, 1}
%t A144406 g[x_, n_] = x^(n) - (x^n - 1)/(x - 1);
%t A144406 h[x_, n_] = FullSimplify[ExpandAll[x^(n)*g[1/x, n]]];
%t A144406 f[t_, n_] := 1/h[t, n];
%t A144406 a = Table[CoefficientList[Series[f[t, m], {t, 0, 30}], t], {m, 1, 31}];
%t A144406 b = Table[Table[a[[n - m + 1]][[m]], {m, 1, n }], {n, 1, 15}];
%t A144406 Flatten[b] (* Triangle version *)
%t A144406 Grid[Table[CoefficientList[Series[(1 - x)/(1 - 2 x + x^(n + 1)), {x, 0, 10}], x], {n, 1, 10}]] (* Array version - _L. Edson Jeffery_, Jul 18 2014 *)
%Y A144406 Cf. A000079, A045623, A092921, A175331.
%Y A144406 Same as A048887 but with a column of 1's added on the left (the number of compositions of 0 is defined to be equal to 1).
%Y A144406 Array rows (with appropriate offsets) are A000012, A000045, A000073, A000078, A001591, A001592, etc.
%K A144406 nonn,tabl
%O A144406 1,9
%A A144406 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 29 2008
%E A144406 Definition changed by _L. Edson Jeffery_, Jul 18 2014