A307796 Number T(n,k) of binary words of length n such that k is the difference of numbers of occurrences of subword 101 and subword 010; triangle T(n,k), n>=0, -floor(n/3)<=k<=floor(n/3), read by rows.

1, 2, 4, 1, 6, 1, 2, 12, 2, 6, 20, 6, 1, 12, 38, 12, 1, 3, 28, 66, 28, 3, 10, 56, 124, 56, 10, 1, 24, 119, 224, 119, 24, 1, 4, 60, 236, 424, 236, 60, 4, 15, 134, 481, 788, 481, 134, 15, 1, 42, 304, 950, 1502, 950, 304, 42, 1, 5, 114, 656, 1902, 2838, 1902, 656, 114, 5

Offset

0,2

Links

Alois P. Heinz, Rows n = 0..250, flattened

Formula

T(n,k) = T(n,-k).

Sum_{k = -floor(n/3)..floor(n/3)} T(n,k) * k^2/2 = A057711(n-2) for n > 1.

Example

T(8,2) = 10: 01101101, 10101101, 10110101, 10110110, 10110111, 10111011, 10111101, 11011011, 11011101, 11101101.
T(8,-2) = 10: 00010010, 00100010, 00100100, 01000010, 01000100, 01001000, 01001001, 01001010, 01010010, 10010010.
T(9,3)  = 1: 101101101.
T(9,-3) = 1: 010010010.
Triangle T(n,k) begins:
  :                      1                   ;
  :                      2                   ;
  :                      4                   ;
  :                1,    6,   1              ;
  :                2,   12,   2              ;
  :                6,   20,   6              ;
  :           1,  12,   38,  12,   1         ;
  :           3,  28,   66,  28,   3         ;
  :          10,  56,  124,  56,  10         ;
  :      1,  24, 119,  224, 119,  24,  1     ;
  :      4,  60, 236,  424, 236,  60,  4     ;
  :     15, 134, 481,  788, 481, 134, 15     ;
  :  1, 42, 304, 950, 1502, 950, 304, 42, 1  ;

Maple

b:= proc(n, t, h) option remember; `if`(n=0, 1, expand(
      `if`(h=3, 1/x, 1)*b(n-1, [1, 3, 1][t], 2)+
      `if`(t=3, x, 1)*b(n-1, 2, [1, 3, 1][h])))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=-iquo(n, 3)..iquo(n, 3)))(b(n, 1$2)):
seq(T(n), n=0..15);

Mathematica

b[n_, t_, h_] := b[n, t, h] = If[n == 0, 1, Expand[If[h == 3, 1/x, 1]* b[n-1, {1, 3, 1}[[t]], 2] + If[t == 3, x, 1]*b[n-1, 2, {1, 3, 1}[[h]]]]];
T[n_] := Table[Coefficient[#, x, i], {i, -Quotient[n, 3], Quotient[n, 3]}]& @ b[n, 1, 1];
Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, May 08 2019, after Alois P. Heinz *)

Crossrefs

Columns k=0-2 give: A164146, A284449, A286209.

Row sums give A000079.

T(3n-4,n-2) gives A000217 for n >= 3.

Cf. A002264, A057711, A303696.

Sequence in context: A047908 A125847 A078886 * A095247 A007734 A171233

Adjacent sequences: A307793 A307794 A307795 * A307797 A307798 A307799

Keyword

nonn,tabf

Author

Alois P. Heinz, Apr 29 2019

Status

approved