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A277751 Number T(n,k) of binary words of length n containing exactly k (possibly overlapping) occurrences of the subword 01101; triangle T(n,k), n>=0, k=0..max(0,floor((n-2)/3)), read by rows.

%I A277751 #19 Feb 22 2021 06:52:21
%S A277751 1,2,4,8,16,31,1,60,4,116,12,225,30,1,436,72,4,845,166,13,1637,375,35,
%T A277751 1,3172,828,92,4,6146,1802,230,14,11909,3872,562,40,1,23075,8243,1333,
%U A277751 113,4,44711,17404,3106,300,15,86633,36501,7114,778,45,1,167863,76104
%N A277751 Number T(n,k) of binary words of length n containing exactly k (possibly overlapping) occurrences of the subword 01101; triangle T(n,k), n>=0, k=0..max(0,floor((n-2)/3)), read by rows.
%H A277751 Alois P. Heinz, <a href="/A277751/b277751.txt">Rows n = 0..350, flattened</a>
%F A277751 G.f. of column k=0: (x+1)*(x^2-x+1)/(x^5-2*x^4+x^3-2*x+1); g.f. of column k>0: x^5*(x^3*(x-1)^2)^(k-1)/(x^5+x^4-x^3+2*x-1)^(k+1).
%F A277751 Sum_{k>=0} k * T(n,k) = A001787(n-4) for n > 3.
%e A277751 Triangle T(n,k) begins:
%e A277751 :     1;
%e A277751 :     2;
%e A277751 :     4;
%e A277751 :     8;
%e A277751 :    16;
%e A277751 :    31,   1;
%e A277751 :    60,   4;
%e A277751 :   116,  12;
%e A277751 :   225,  30,  1;
%e A277751 :   436,  72,  4;
%e A277751 :   845, 166, 13;
%e A277751 :  1637, 375, 35, 1;
%e A277751 :  3172, 828, 92, 4;
%p A277751 b:= proc(n, t) option remember; expand(
%p A277751       `if`(n=0, 1,     b(n-1, [2, 2, 2, 5, 2][t])+
%p A277751       `if`(t=5, x, 1)* b(n-1, [1, 3, 4, 1, 3][t])))
%p A277751     end:
%p A277751 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 1)):
%p A277751 seq(T(n), n=0..20);
%p A277751 # second Maple program:
%p A277751 gf:= k-> `if`(k=0, (x+1)*(x^2-x+1), x^5*(x^3*(x-1)^2)^(k-1))
%p A277751                    /(x^5-2*x^4+x^3-2*x+1)^(k+1):
%p A277751 T:= (n, k)-> coeff(series(gf(k), x, n+1), x, n):
%p A277751 seq(seq(T(n, k), k=0..max(0, floor((n-2)/3))), n=0..20);
%t A277751 b[n_, t_] := b[n, t] = Expand[
%t A277751      If[n==0, 1,     b[n-1, {2, 2, 2, 5, 2}[[t]]] +
%t A277751      If[t==5, x, 1]* b[n-1, {1, 3, 4, 1, 3}[[t]]]]];
%t A277751 T[n_] := CoefficientList[b[n, 1], x];
%t A277751 T /@ Range[0, 20] // Flatten (* _Jean-François Alcover_, Feb 22 2021, after first Maple program *)
%Y A277751 Columns k=0-2 give: A209888, A317780, A317781.
%Y A277751 Row sums give A000079.
%Y A277751 Cf. A001787, A317669.
%K A277751 nonn,tabf
%O A277751 0,2
%A A277751 _Alois P. Heinz_, Oct 28 2016