|
%I #32 Oct 13 2022 05:56:25
%S 0,1,8,105,6136,8766473,8245941529080,3508518207951157937469961,
%T 311594265746788494170062926869662848646207622648,
%U 1217308491239906829392988008143949647398943617188660186130545502913055217344025410733271773705
%N Sum of the sizes of binary subtrees of the perfect binary tree of height n.
%C Size of binary tree = number of internal nodes.
%H Alois P. Heinz, <a href="/A024358/b024358.txt">Table of n, a(n) for n = 0..12</a>
%H Cyril Banderier, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/subtrees.pdf">On the sum of the sizes of binary subtrees of a perfect binary tree</a>, personal note, 2000.
%F a(n) = B'_n(1) where B_{n+1}(x) = 1 + x*B_n(x)^2.
%F From _Alois P. Heinz_, Jul 12 2019: (Start)
%F a(n) = Sum_{k=0..2^n-1} (2^n-1-k) * A309049(2^n-1,k).
%F a(n) = A309052(2^n-1). (End)
%p B:= proc(n) B(n):= `if`(n<0, 0, expand(1+x*B(n-1)^2)) end:
%p a:= n-> subs(x=1, diff(B(n), x)):
%p seq(a(n), n=0..9); # _Alois P. Heinz_, Jul 12 2019
%t B[n_] := If[n<0, 0, Expand[1+x*B[n-1]^2]];
%t a[n_] := D[B[n], x] /. x -> 1;
%t Table[a[n], {n, 0, 9}] (* _Jean-François Alcover_, Oct 13 2022, after _Alois P. Heinz_ *)
%Y Cf. A003095, A309049, A309052.
%K easy,nonn
%O 0,3
%A _Cyril Banderier_, Jun 09 2000
%E a(0) changed to 0 by _Alois P. Heinz_, Jul 12 2019
| 