A309051 Total number of 0's in all (binary) max-heaps on n elements from the set {0,1}.

0, 1, 3, 7, 13, 24, 42, 77, 122, 206, 332, 578, 889, 1484, 2338, 4019, 5960, 9685, 14887, 25134, 37225, 60704, 92919, 156646, 227302, 364551, 550329, 917822, 1337358, 2158150, 3258779, 5441757, 7800755, 12412461, 18546566, 30708486, 44327782, 71090442

Offset: 0

Author

Alois P. Heinz, Jul 09 2019

Keywords

• nonn

Comments

Also the total number of 1's in all (binary) min-heaps on n elements from the set {0,1}.

Examples

			a(4) = 13 = 4+3+2+2+1+1+0, the total number of 0's in 0000, 1000, 1010, 1100, 1101, 1110, 1111.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5632
• Eric Weisstein's World of Mathematics, Heap
• Wikipedia, Binary heap

Crossrefs

Cf. A056971, A091980, A309049, A309052.

Programs

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:
a:= n-> subs(x=1, diff(b(n), x)):
seq(a(n), n=0..40);

Mathematica

b[n_][x_] := b[n][x] = If[n == 0, 1, Function[g, Function[f, Expand[x^n + b[f][x] b[n - 1 - f][x]]][Min[g - 1, n - g/2]]][2^(Length[IntegerDigits[ n, 2]] - 1)]];
a[n_] := b[n]'[1];
a /@ Range[0, 40] (* Jean-François Alcover, Apr 22 2021, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..n} k * A309049(n,k).

a(n) = n * A091980(n+1) - A309052(n).