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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-Fran├žois 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 September 25 02:25 EDT 2021. Contains 347651 sequences. (Running on oeis4.)