A372641 - OEIS

%I #14 May 09 2024 15:55:53

%S 1,0,0,0,0,1,2,2,9,11,36,71,151,306,591,1228,2469,4966,10025,19591,

%T 38946,75977,148585,291027,579981,1152385,2280696,4470814,8817933,

%U 17244969,33819425,65976444,129933731,254791662,499516984,977417823,1914394157,3745482924

%N Number of defective (binary) heaps on n elements from the set {0,1} where exactly n ancestor-successor pairs do not have the correct order.

%H Alois P. Heinz, <a href="/A372641/b372641.txt">Table of n, a(n) for n = 0..2000</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) = A372640(n,n).

%e a(5) = 1: 00111.

%e a(6) = 2: 000111, 001111.

%e a(7) = 2: 0011111, 0101111.

%e a(8) = 9: 00010111, 00011011, 00011101, 00011110, 00101111, 00111011, 00111101, 01001111, 01011111.

%e a(9) = 11: 000011101, 000011110, 001001111, 001010011, 001101111, 001110011, 001111101, 001111110, 010001111, 010111111, 011011111.

%e (The examples use max-heaps.)

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

%p       expand(b(f, t)*b(n-1-f, t)*x^t+b(f, t+1)*b(n-1-f, t+1)

%p           ))(min(g-1, n-g/2)))(2^ilog2(n)))

%p     end:

%p a:= n-> coeff(b(n, 0),x,n):

%p seq(a(n), n=0..37);

%t b[n_, t_] := b[n, t] = If[n == 0, 1, Function[g, Function [f,

%t    Expand[b[f, t]*b[n-1-f, t]*x^t + b[f, t+1]*b[n-1-f, t+1]]][

%t    Min[g-1, n-g/2]]][2^(Length@IntegerDigits[n, 2]-1)]];

%t a[n_] := Coefficient[b[n, 0], x, n];

%t Table[a[n], {n, 0, 37}] (* _Jean-François Alcover_, May 09 2024, after _Alois P. Heinz_ *)

%Y Main diagonal of A372640.

%K nonn

%O 0,7

%A _Alois P. Heinz_, May 08 2024