A386912 The base-2 logarithm of this sequence equals the Q statistic value of the greedy-from-the-bottom tree with n leaves, which is the minimal Q statistic value for n.

1, 2, 6, 16, 60, 192, 672, 2048, 8640, 30720, 118272, 393216, 1597440, 5505024, 20643840, 67108864, 300810240, 1132462080, 4602200064, 16106127360, 68702699520, 248034361344, 972407439360, 3298534883328, 14495514624000, 53601191854080

Offset: 1

Author

Mareike Fischer, Aug 07 2025

Keywords

• nonn

Examples

			a(1)=1, because a(1)=Product(i=2..1)=1 (empty product).
a(2)=2, because q(2,2)=1, so a(2)=2^q(2,2)=2^1=2.
a(3)=6, because q(3,2)=1 and q(3,3)=1, so a(3)=2^q(3,2)*3^q(3,3)=2^1*3^1=6.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1666 (first 300 terms from Mareike Fischer)
• Sean Cleary, Mareike Fischer, and Katherine St. John, The GFB Tree and Tree Imbalance Indices, arXiv:2502.12854 [q-bio.PE], 2025.

Crossrefs

Cf. A132862.

Programs

Maple

a:= proc(n) option remember; `if`(n<2, 1, (b-> (f->
      a(f)*n*a(n-f))(min(b, n-b/2)))(2^ilog2(n-1)))
    end:
seq(a(n), n=1..26);  # Alois P. Heinz, Aug 18 2025

Mathematica

q[n_, i_] := Module[{factor, ki}, factor = FactorInteger[i];
  ki = Ceiling[Log2[i]];
  If[(i == 1 || i == n), Return[n/i],
   If[(Length[factor] == 1 && factor[[1]][[1]] == 2),
     If[(Mod[n, i] == 0 || Mod[n, i] >= 2^(ki - 1)),
      Return[Floor[n/i]], Return[Floor[n/i] - 1]],
     If[Mod[n - i, 2^(ki - 1)] == 0, Return[1], Return[0]]];
   ]];
a[n_] := Product[i^q[n, i], {i, 2, n}]; alist = {};
For[n = 1, n <= 300, n++, AppendTo[alist, a[n]]]; Print[alist]
(* Alternative, after Alois P. Heinz: *)
A386912[n_] := A386912[n] = If[n == 1, 1, n*A386912[#]*A386912[n - #]] & [Min[#, n - #/2] & [2^(BitLength[n - 1] - 1)]]; Array[A386912, 30] (* Paolo Xausa, Aug 19 2025 *)

Python

from functools import lru_cache
@lru_cache(maxsize=None)
def a(n: int) -> int:
    if n < 2: return 1
    b = 1 << ((n - 1).bit_length() - 1)
    f = min(b, n - b // 2)
    return a(f) * n * a(n - f)
print([a(n) for n in range(1, 27)])  # after Alois P. Heinz, Peter Luschny, Aug 19 2025

Formula

a(n) = Product_{i=2..n} i^q(n,i), with k(i)=ceiling(log_2(i)) and q(n,i)=floor(n/i) if (i=2^(k(i)) and ((n mod i)=0 or (n mod i)>=2^(k(i)-1))), and q(n,i)=floor(n/i)-1 if (i=2^(k(i)) and (0 < (n mod i)< 2^(k(i)-1))) and q(n,i)=1 if (i!=2^(k(i)) and ( (n-i) mod 2^(k(i)-1)) =0 ) and q(n,i)=0 if (i!=2^(k(i)) and ( (n-i) mod 2^(k(i)-1)) >0 ).

a(n) = Product_{i=0..k(n)-1} (2^(k(n)-i))^(2^i) if d(n)=2^(k(n)-1). Else, a(n) = Product_{i=0..k(n)-2} (2^(k(n)-i-1))^(2^i) * 2^d(n) * Product_{i=1..k(n)-1} ( 2^floor(d(n)/(2^i)) * ((2^i + (d(n)-2^i*(floor(d(n)/(2^i))))) / (2^i) )^(ceiling(d(n)/2^i)-floor(d(n)/(2^i))) if d(n) < 2^(k(n)-1), with k(n)=ceiling(log_2(n)) and d(n)=n-2^(k(n)-1).

a(n) = n*a(f)*a(n-f) with f = min(b,n-b/2) and b = 2^floor(log_2(n-1)) for n>1, a(1) = 1. - Alois P. Heinz, Aug 18 2025