A309049 Number T(n,k) of (binary) max-heaps on n elements from the set {0,1} containing exactly k 0's; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 3, 4, 4, 2, 1, 1, 1, 4, 6, 6, 5, 2, 1, 1, 1, 4, 7, 8, 7, 5, 2, 1, 1, 1, 5, 10, 12, 11, 8, 5, 2, 1, 1, 1, 5, 11, 16, 17, 13, 9, 5, 2, 1, 1, 1, 6, 15, 23, 27, 24, 16, 10, 5, 2, 1, 1, 1, 6, 16, 27, 34, 34, 27, 18, 11, 5, 2, 1, 1

Offset

0,8

Comments

Also the number T(n,k) of (binary) min-heaps on n elements from the set {0,1} containing exactly k 1's.

The sequence of column k satisfies a linear recurrence with constant coefficients of order A063915(k).

Links

Alois P. Heinz, Rows n = 0..200, flattened

Eric Weisstein's World of Mathematics, Heap

Wikipedia, Binary heap

Formula

Sum_{k=0..n} k * T(n,k) = A309051(n).

Sum_{k=0..n} (n-k) * T(n,k) = A309052(n).

Sum_{k=0..2^n-1} T(2^n-1,k) = A003095(n+1).

Sum_{k=0..2^n-1} (2^n-1-k) * T(2^n-1,k) = A024358(n).

Sum_{k=0..n} (T(n,k) - T(n-1,k)) = A168542(n).

T(m,m-n) = A000108(n) for m >= 2^n-1 = A000225(n).

T(2^n-1,k) = A202019(n+1,k+1).

Example

T(6,0) = 1: 111111.
T(6,1) = 3: 111011, 111101, 111110.
T(6,2) = 4: 110110, 111001, 111010, 111100.
T(6,3) = 4: 101001, 110010, 110100, 111000.
T(6,4) = 2: 101000, 110000.
T(6,5) = 1: 100000.
T(6,6) = 1: 000000.
T(7,4) = T(7,7-3) = A000108(3) = 5: 1010001, 1010010, 1100100, 1101000, 1110000.
Triangle T(n,k) begins:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  1,  1;
  1, 2,  2,  1,  1;
  1, 3,  3,  2,  1,  1;
  1, 3,  4,  4,  2,  1,  1;
  1, 4,  6,  6,  5,  2,  1,  1;
  1, 4,  7,  8,  7,  5,  2,  1,  1;
  1, 5, 10, 12, 11,  8,  5,  2,  1, 1;
  1, 5, 11, 16, 17, 13,  9,  5,  2, 1, 1;
  1, 6, 15, 23, 27, 24, 16, 10,  5, 2, 1, 1;
  1, 6, 16, 27, 34, 34, 27, 18, 11, 5, 2, 1, 1;
  ...

Maple

b:= proc(n) option remember; `if`(n=0, 1, (g-> (f-> expand(
      x^n+b(f)*b(n-1-f)))(min(g-1, n-g/2)))(2^ilog2(n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..14);

Mathematica

b[n_] := b[n] = If[n == 0, 1, Function[g, Function[f, Expand[x^n + b[f]*b[n - 1 - f]]][Min[g - 1, n - g/2]]][2^Floor[Log[2, n]]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n]];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Oct 04 2019, after Alois P. Heinz *)

Crossrefs

Columns k=0-10 give: A000012, A110654, A114220 (for n>0), A326504, A326505, A326506, A326507, A326508, A326509, A326510, A326511.

Row sums give A091980(n+1).

T(2n,n) gives A309050.

Rows reversed converge to A000108.

Cf. A000225, A000295, A003095, A024358, A056971, A063915, A137560, A168542, A202019, A309051, A309052.

Sequence in context: A008284 A114088 A208245 * A274190 A322596 A037306

Adjacent sequences: A309046 A309047 A309048 * A309050 A309051 A309052

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 09 2019

Status

approved