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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. 5
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified July 12 21:31 EDT 2020. Contains 335669 sequences. (Running on oeis4.)