%I #45 Jun 04 2024 16:54:15

%S 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 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 61,108,149,89,1,1,2,4,8,16,32,63,120,208,274,144,1

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

%C 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 T(m,n) is the number of binary words of length n-1 with < m consecutive 1's. - _Geoffrey Critzer_, Sep 02 2012

%D J. Riordan, An Introduction to Combinatorial Analysis, Princeton University Press, Princeton, NJ, 1978, p. 154.

%H Alois P. Heinz, <a href="/A048887/b048887.txt">Antidiagonals n = 1..141, flattened</a>

%H 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 G.f.: (1-z)/[1-2z+z^(t+1)].

%e T(2,5) counts 11111, 1112, 1121, 1211, 2111, 122, 212, 221, where "1211" abbreviates the composition 1+2+1+1.

%e 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 The array begins:

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 2, 3, 5, 8, 13, ...

%e 1, 2, 4, 7, 13, ...

%e 1, 2, 4, 8, ...

%e 1, 2, 4, ...

%e 1, 2, ...

%e 1, ...

%p 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 # second Maple program:

%p T:= proc(m, n) option remember; `if`(n=0 or m=1, 1,

%p add(T(m, n-j), j=1..min(n, m)))

%p end:

%p seq(seq(T(1+d-n, n), n=1..d), d=1..14); # _Alois P. Heinz_, May 21 2013

%t 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 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 Rows: A000045 (Fibonacci), A000073 (tribonacci), A000078 (tetranacci), etc.

%Y Essentially a reflected version of A092921. See A048004 and A126198 for closely related arrays.

%K nonn,tabl

%O 1,5

%A _Clark Kimberling_