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A277678 Number T(n,k) of binary words of length n containing exactly k (possibly overlapping) occurrences of the subword 11011; triangle T(n,k), n>=0, k=0..max(0,floor((n-2)/3)), read by rows.

%I A277678 #18 Apr 29 2022 12:14:22
%S A277678 1,2,4,8,16,31,1,60,4,116,12,225,30,1,437,70,5,849,158,17,1649,351,47,
%T A277678 1,3202,770,118,6,6217,1669,283,23,12071,3578,664,70,1,23438,7599,
%U A277678 1535,189,7,45510,16016,3500,480,30,88368,33545,7876,1182,100,1,171586
%N A277678 Number T(n,k) of binary words of length n containing exactly k (possibly overlapping) occurrences of the subword 11011; triangle T(n,k), n>=0, k=0..max(0,floor((n-2)/3)), read by rows.
%H A277678 Alois P. Heinz, <a href="/A277678/b277678.txt">Rows n = 0..350, flattened</a>
%F A277678 G.f. of column k=0: -(x^4+x^3+1)/(x^5+x^4-x^3+2*x-1); g.f. of column k>0: x^5*(x^3*(x^2+x-1))^(k-1)/(x^5+x^4-x^3+2*x-1)^(k+1).
%F A277678 Sum_{k>=0} k * T(n,k) = A001787(n-4) for n>3.
%e A277678 Triangle T(n,k) begins:
%e A277678 :     1;
%e A277678 :     2;
%e A277678 :     4;
%e A277678 :     8;
%e A277678 :    16;
%e A277678 :    31,   1;
%e A277678 :    60,   4;
%e A277678 :   116,  12;
%e A277678 :   225,  30,   1;
%e A277678 :   437,  70,   5;
%e A277678 :   849, 158,  17;
%e A277678 :  1649, 351,  47, 1;
%e A277678 :  3202, 770, 118, 6;
%p A277678 b:= proc(n, t) option remember; expand(
%p A277678       `if`(n=0, 1,     b(n-1, [1, 1, 4, 1, 1][t])+
%p A277678       `if`(t=5, x, 1)* b(n-1, [2, 3, 3, 5, 3][t])))
%p A277678     end:
%p A277678 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 1)):
%p A277678 seq(T(n), n=0..20);
%p A277678 # second Maple program:
%p A277678 gf:= k-> `if`(k=0, -(x^4+x^3+1), x^5*(x^3*(x^2+x-1))^(k-1))
%p A277678                    /(x^5+x^4-x^3+2*x-1)^(k+1):
%p A277678 T:= (n, k)-> coeff(series(gf(k), x, n+1), x, n):
%p A277678 seq(seq(T(n, k), k=0..max(0, floor((n-2)/3))), n=0..20);
%t A277678 b[n_, t_] := b[n, t] = Expand[
%t A277678      If[n == 0, 1,    b[n-1, {1, 1, 4, 1, 1}[[t]]] +
%t A277678      If[t == 5, x, 1]*b[n-1, {2, 3, 3, 5, 3}[[t]]]]];
%t A277678 T[n_] := CoefficientList[b[n, 1], x];
%t A277678 Table[T[n], {n, 0, 20}] // Flatten (* _Jean-François Alcover_, Apr 29 2022, after _Alois P. Heinz_ *)
%Y A277678 Column k=0 gives A210021.
%Y A277678 Row sums give A000079.
%Y A277678 Row sums except column k=0 give A276785.
%Y A277678 Cf. A001787, A002264.
%K A277678 nonn,tabf
%O A277678 0,2
%A A277678 _Alois P. Heinz_, Oct 26 2016