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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. 5
 1, 2, 4, 8, 16, 31, 1, 60, 4, 116, 12, 225, 30, 1, 436, 72, 4, 845, 166, 13, 1637, 375, 35, 1, 3172, 828, 92, 4, 6146, 1802, 230, 14, 11909, 3872, 562, 40, 1, 23075, 8243, 1333, 113, 4, 44711, 17404, 3106, 300, 15, 86633, 36501, 7114, 778, 45, 1, 167863, 76104 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Alois P. Heinz, Rows n = 0..350, flattened FORMULA 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). Sum_{k>=0} k * T(n,k) = A001787(n-4) for n > 3. EXAMPLE Triangle T(n,k) begins: : 1; : 2; : 4; : 8; : 16; : 31, 1; : 60, 4; : 116, 12; : 225, 30, 1; : 436, 72, 4; : 845, 166, 13; : 1637, 375, 35, 1; : 3172, 828, 92, 4; MAPLE b:= proc(n, t) option remember; expand( `if`(n=0, 1, b(n-1, [2, 2, 2, 5, 2][t])+ `if`(t=5, x, 1)* b(n-1, [1, 3, 4, 1, 3][t]))) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 1)): seq(T(n), n=0..20); # second Maple program: gf:= k-> `if`(k=0, (x+1)*(x^2-x+1), x^5*(x^3*(x-1)^2)^(k-1)) /(x^5-2*x^4+x^3-2*x+1)^(k+1): T:= (n, k)-> coeff(series(gf(k), x, n+1), x, n): seq(seq(T(n, k), k=0..max(0, floor((n-2)/3))), n=0..20); MATHEMATICA b[n_, t_] := b[n, t] = Expand[ If[n==0, 1, b[n-1, {2, 2, 2, 5, 2}[[t]]] + If[t==5, x, 1]* b[n-1, {1, 3, 4, 1, 3}[[t]]]]]; T[n_] := CoefficientList[b[n, 1], x]; T /@ Range[0, 20] // Flatten (* Jean-François Alcover, Feb 22 2021, after first Maple program *) CROSSREFS Columns k=0-2 give: A209888, A317780, A317781. Row sums give A000079. Cf. A001787, A317669. Sequence in context: A283836 A244825 A220843 * A277678 A018763 A054517 Adjacent sequences: A277748 A277749 A277750 * A277752 A277753 A277754 KEYWORD nonn,tabf AUTHOR Alois P. Heinz, Oct 28 2016 STATUS approved

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Last modified August 9 21:53 EDT 2024. Contains 375044 sequences. (Running on oeis4.)