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A145111 Square array A(n,k) of numbers of length n binary words with fewer than k 0-digits between any pair of consecutive 1-digits (n,k >= 0), read by antidiagonals.

%I A145111 #37 Oct 04 2018 20:32:06
%S A145111 1,1,2,1,2,3,1,2,4,4,1,2,4,7,5,1,2,4,8,11,6,1,2,4,8,15,16,7,1,2,4,8,
%T A145111 16,27,22,8,1,2,4,8,16,31,47,29,9,1,2,4,8,16,32,59,80,37,10,1,2,4,8,
%U A145111 16,32,63,111,134,46,11,1,2,4,8,16,32,64,123,207,222,56,12,1,2,4,8,16,32,64,127,239,384,365,67,13
%N A145111 Square array A(n,k) of numbers of length n binary words with fewer than k 0-digits between any pair of consecutive 1-digits (n,k >= 0), read by antidiagonals.
%H A145111 Alois P. Heinz, <a href="/A145111/b145111.txt">Antidiagonals n = 0..140</a>
%F A145111 G.f. of column k: (1-x+x^(k+1))/(1-3*x+2*x^2+x^(k+1)-x^(k+2)).
%e A145111 A(4,1) = 11, because 11 binary words of length 4 have fewer than 1 0-digit between any pair of consecutive 1-digits: 0000, 0001, 0010, 0100, 1000, 0011, 0110, 1100, 0111, 1110, 1111.
%e A145111 Square array A(n,k) begins:
%e A145111   1,  1,  1,  1,  1,  1, ...
%e A145111   2,  2,  2,  2,  2,  2, ...
%e A145111   3,  4,  4,  4,  4,  4, ...
%e A145111   4,  7,  8,  8,  8,  8, ...
%e A145111   5, 11, 15, 16, 16, 16, ...
%e A145111   6, 16, 27, 31, 32, 32, ...
%p A145111 f:= proc(n,k) option remember; local j; if n=0 then 1 elif n<=k then 2^(n-1) else add(f(n-j, k), j=1..k) fi end: g:= proc(n,k) option remember; if n<0 then 0 else g(n-1,k) +f(n,k) fi end: A:= (n,k)-> `if`(n=0, g(0,k), A(n-1,k) +g(n-1,k)): seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A145111 a[n_, k_] := SeriesCoefficient[(1 - x + x^(k+1))/(1 - 3*x + 2*x^2 + x^(k+1) - x^(k+2)), {x, 0, n}]; a[0, _] = 1; Table[a[n-k, k], {n, 0, 14}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Jan 15 2014 *)
%Y A145111 Columns 0-9 give: A000027(n+1), A000124, A000126(n+1), A007800(n+1), A145112, A145113, A145114, A145115, A145116, A145117.
%Y A145111 Main diagonal gives A000079.
%Y A145111 Cf. A141539.
%K A145111 nonn,tabl
%O A145111 0,3
%A A145111 _Alois P. Heinz_, Oct 02 2008