A203303 Vandermonde determinant of the first n terms of (1,2,4,8,16,...).

1, 1, 6, 1008, 20321280, 203199794380800, 4096245678214226116608000, 671169825411994707343327912777482240000, 3589459026274030507466469204160461571257625328222208000000, 2511229721141086754031154605327661795863172723306019839389105937236728217600000000

Offset

1,3

Comments

Each term divides its successor, as in A002884. Indeed, 2*v(n+1)/v(n) divides v(n+2)/v(n+1), as in A171499.

Links

Robert Israel, Table of n, a(n) for n = 1..22

Daniel M. Kane, Carlo Sanna, and Jeffrey Shallit, Waring's theorem for binary powers, arXiv:1801.04483 [math.NT], Jan 13 2018.

Formula

a(n) = Product_{0 <= i < j <= n-1} (2^j - 2^i) = 2^(n*(n-1)*(n-2)/6) * Product_{1<=k<=n-1} (2^k-1)^(n-k). - Robert Israel, Jan 16 2018

a(n) ~ 1/A335011 * 2^(n*(n-1)*(2*n-1)/6) * QPochhammer(1/2)^n. - Vaclav Kotesovec, May 19 2020

Maple

with(LinearAlgebra):
a:= n-> Determinant(VandermondeMatrix([2^i$i=0..n-1])):
seq(a(n), n=1..12);  # Alois P. Heinz, Jul 23 2017
f:= n -> 2^(n*(n-1)*(n-2)/6)*mul((2^k-1)^(n-k), k=1..n-1):
seq(f(n), n=1..12); # Robert Israel, Jan 16 2018

Mathematica

f[j_] := 2^(j - 1); z = 15;
v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]   (* A203303 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]  (* A002884 *)
Table[v[n] v[n + 2]/(2*v[n + 1]^2), {n, 1, z - 1}]  (* A171499 *)
Table[FactorInteger[v[n]], {n, 1, z - 1}]

Crossrefs

Cf. A000079.

Sequence in context: A024085 A080474 A079190 * A159865 A004806 A282233

Adjacent sequences: A203300 A203301 A203302 * A203304 A203305 A203306

Keyword

nonn

Author

Clark Kimberling, Jan 01 2012

Status

approved