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A048887 Array T read by antidiagonals, where T(m,n) = number of compositions of n into parts <= m.

%I A048887 #45 Jun 04 2024 16:54:15
%S A048887 1,1,1,1,2,1,1,2,3,1,1,2,4,5,1,1,2,4,7,8,1,1,2,4,8,13,13,1,1,2,4,8,15,
%T A048887 24,21,1,1,2,4,8,16,29,44,34,1,1,2,4,8,16,31,56,81,55,1,1,2,4,8,16,32,
%U A048887 61,108,149,89,1,1,2,4,8,16,32,63,120,208,274,144,1
%N A048887 Array T read by antidiagonals, where T(m,n) = number of compositions of n into parts <= m.
%C A048887 Taking finite differences of array columns from the top down, we obtain (1; 1,1; 1,2,1; 1,4,2,1; ...) = A048004 rows. - _Gary W. Adamson_, Aug 20 2010
%C A048887 T(m,n) is the number of binary words of length n-1 with < m consecutive 1's. - _Geoffrey Critzer_, Sep 02 2012
%D A048887 J. Riordan, An Introduction to Combinatorial Analysis, Princeton University Press, Princeton, NJ, 1978, p. 154.
%H A048887 Alois P. Heinz, <a href="/A048887/b048887.txt">Antidiagonals n = 1..141, flattened</a>
%H A048887 Hsin-Po Wang and Chi-Wei Chin, <a href="https://arxiv.org/abs/2405.17499">On Counting Subsequences and Higher-Order Fibonacci Numbers</a>, arXiv:2405.17499 [cs.IT], 2024. See p. 2.
%F A048887 G.f.: (1-z)/[1-2z+z^(t+1)].
%e A048887 T(2,5) counts 11111, 1112, 1121, 1211, 2111, 122, 212, 221, where "1211" abbreviates the composition 1+2+1+1.
%e A048887 These eight compositions correspond respectively to: {0,0,0,0}, {0,0,0,1}, {0,0,1,0}, {0,1,0,0}, {1,0,0,0}, {0,1,0,1}, {1,0,0,1}, {1,0,1,0} per the bijection given by _N. J. A. Sloane_ in A048004. - _Geoffrey Critzer_, Sep 02 2012
%e A048887 The array begins:
%e A048887   1,  1,  1,  1,  1,  1,  1, ...
%e A048887   1,  2,  3,  5,  8, 13, ...
%e A048887   1,  2,  4,  7, 13, ...
%e A048887   1,  2,  4,  8, ...
%e A048887   1,  2,  4, ...
%e A048887   1,  2, ...
%e A048887   1, ...
%p A048887 G := t->(1-z)/(1-2*z+z^(t+1)): T := (m,n)->coeff(series(G(m),z=0,30),z^n): matrix(7,12,T);
%p A048887 # second Maple program:
%p A048887 T:= proc(m, n) option remember; `if`(n=0 or m=1, 1,
%p A048887       add(T(m, n-j), j=1..min(n, m)))
%p A048887     end:
%p A048887 seq(seq(T(1+d-n, n), n=1..d), d=1..14); # _Alois P. Heinz_, May 21 2013
%t A048887 Table[nn=10;a=(1-x^k)/(1-x);b=1/(1-x);c=(1-x^(k-1))/(1-x); CoefficientList[ Series[a b/(1-x^2 b c), {x,0,nn}],x],{k,1,nn}]//Grid  (* _Geoffrey Critzer_, Sep 02 2012 *)
%t A048887 T[m_, n_] := T[m, n] = If[n == 0 || m == 1, 1, Sum[T[m, n-j], {j, 1, Min[n, m]}]]; Table[Table[T[1+d-n, n], {n, 1, d}], {d, 1, 14}] // Flatten (* _Jean-François Alcover_, Nov 12 2014, after _Alois P. Heinz_ *)
%Y A048887 Rows: A000045 (Fibonacci), A000073 (tribonacci), A000078 (tetranacci), etc.
%Y A048887 Essentially a reflected version of A092921. See A048004 and A126198 for closely related arrays.
%K A048887 nonn,tabl
%O A048887 1,5
%A A048887 _Clark Kimberling_