A309052 - OEIS

%I #29 Apr 22 2021 05:24:47

%S 0,1,3,8,15,31,54,105,166,298,478,863,1307,2247,3500,6136,9032,15084,

%T 23039,39599,57955,96019,145627,248223,357650,583274,875459,1476754,

%U 2131618,3476550,5210521,8766473,12498445,20138409,29952394,50020414,71658602,115850282

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

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

%H Alois P. Heinz, <a href="/A309052/b309052.txt">Table of n, a(n) for n = 0..5631</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Heap.html">Heap</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Binary_heap">Binary heap</a>

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

%F a(n) = n * A091980(n+1) - A309051(n).

%F a(2^n-1) = A024358(n).

%e a(4) = 15 = 0+1+2+2+3+3+4, the total number of 1's in 0000, 1000, 1010, 1100, 1101, 1110, 1111.

%p b:= proc(n) option remember; `if`(n=0, 1, (g-> (f-> expand(

%p       1+x*b(f)*b(n-1-f)))(min(g-1, n-g/2)))(2^ilog2(n)))

%p     end:

%p a:= n-> subs(x=1, diff(b(n), x)):

%p seq(a(n), n=0..40);

%t b[n_][x_] := b[n][x] = If[n == 0, 1, Function[g, Function[f, Expand[1 + x b[f][x] b[n - 1 - f][x]]][Min[g - 1, n - g/2]]][2^(Length[IntegerDigits[ n, 2]] - 1)]];

%t a[n_] := b[n]'[1];

%t a /@ Range[0, 40] (* _Jean-François Alcover_, Apr 22 2021, after _Alois P. Heinz_ *)

%Y Cf. A024358, A056971, A091980, A309049, A309051.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jul 09 2019