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A232435 Number T(n,k) of compositions of n with exactly k (possibly overlapping) occurrences of the consecutive pattern 111; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.

%I A232435 #11 Feb 08 2017 11:41:02
%S A232435 1,1,2,3,1,7,0,1,13,2,0,1,24,5,2,0,1,46,11,4,2,0,1,89,21,11,4,2,0,1,
%T A232435 170,45,23,11,4,2,0,1,324,99,47,23,12,4,2,0,1,618,209,102,52,23,13,4,
%U A232435 2,0,1,1183,427,226,112,55,24,14,4,2,0,1,2260,883,479
%N A232435 Number T(n,k) of compositions of n with exactly k (possibly overlapping) occurrences of the consecutive pattern 111; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.
%H A232435 Alois P. Heinz, <a href="/A232435/b232435.txt">Rows n = 0..150, flattened</a>
%e A232435 T(4,0) = 7: [4], [3,1], [2,2], [1,3], [2,1,1], [1,2,1], [1,1,2].
%e A232435 T(7,1) = 11: [4,1,1,1], [2,2,2,1], [1,2,2,2], [1,1,1,4], [1,3,1,1,1], [2,2,1,1,1], [1,1,1,3,1], [2,1,1,1,2], [1,1,1,2,2], [1,1,1,2,1,1], [1,1,2,1,1,1].
%e A232435 T(7,2) = 4: [3,1,1,1,1], [1,1,1,1,3], [1,2,1,1,1,1], [1,1,1,1,2,1].
%e A232435 T(7,3) = 2: [2,1,1,1,1,1], [1,1,1,1,1,2].
%e A232435 T(7,5) = 1: [1,1,1,1,1,1,1].
%e A232435 Triangle T(n,k) begins:
%e A232435 :  0 :   1;
%e A232435 :  1 :   1;
%e A232435 :  2 :   2;
%e A232435 :  3 :   3,  1;
%e A232435 :  4 :   7,  0,  1;
%e A232435 :  5 :  13,  2,  0,  1;
%e A232435 :  6 :  24,  5,  2,  0,  1;
%e A232435 :  7 :  46, 11,  4,  2,  0, 1;
%e A232435 :  8 :  89, 21, 11,  4,  2, 0, 1;
%e A232435 :  9 : 170, 45, 23, 11,  4, 2, 0, 1;
%e A232435 : 10 : 324, 99, 47, 23, 12, 4, 2, 0, 1;
%p A232435 b:= proc(n, t) option remember; `if`(n=0, 1,
%p A232435       expand(add(`if`(abs(t)<>j, b(n-j, j),
%p A232435       `if`(t<0, x, 1)*b(n-j, -j)), j=1..n)))
%p A232435     end:
%p A232435 T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
%p A232435 seq(T(n), n=0..15);
%t A232435 b[n_, t_] := b[n, t] = If[n==0, 1, Expand[Sum[If[Abs[t] != j, b[n-j, j], If[t<0, x, 1]*b[n-j, -j]], {j, 1, n}]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 15}] // Flatten (* _Jean-François Alcover_, Feb 08 2017, translated from Maple *)
%Y A232435 Column k=0 gives: A128695.
%Y A232435 Row sums give: A011782.
%K A232435 nonn,tabf
%O A232435 0,3
%A A232435 _Alois P. Heinz_, Nov 23 2013