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

From Robert Israel, Jan 16 2018: (Start)

a(n) = Product_{0 <= i < j <= n-1} (2^j - 2^i).

a(n) = 2^(n*(n-1)*(n-2)/6) * Product_{1<=k<=n-1} (2^k-1)^(n-k). (End)

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

a(n) = Product_{k=0..n-2} ( 2^(k+1)^2 * QPochhammer(2^(-k-1); 2; k+1) ). - G. C. Greubel, Aug 31 2023

Maple

# First program
with(LinearAlgebra):
a:= n-> Determinant(VandermondeMatrix([2^i$i=0..n-1])):
seq(a(n), n=1..12);  # Alois P. Heinz, Jul 23 2017
# Second program
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

(* First program *)
f[j_]:= 2^(j-1); z = 15;
v[n_]:= Product[Product[f[k] - f[j], {j, k-1}], {k, 2, n}]
Table[v[n], {n, z}]                     (* A203303 *)
Table[v[n+1]/v[n], {n, z}]              (* A002884 *)
Table[v[n]*v[n+2]/(2*v[n+1]^2), {n, z}]  (* A171499 *)
Table[FactorInteger[v[n]], {n, z}]
(* Second program *)
Table[Product[2^(k+1) -2^j, {k, 0, n-2}, {j, 0, k}], {n, 15}] (* G. C. Greubel, Aug 31 2023 *)

Prog

(Magma) [1] cat [(&*[(&*[2^(k+1) -2^j: j in [0..k]]): k in [0..n-2]]): n in [2..15]]; // G. C. Greubel, Aug 31 2023
(SageMath) [product(product(2^(k+1) -2^j for j in range(k+1)) for k in range(n-1)) for n in range(1, 16)] # G. C. Greubel, Aug 31 2023

Crossrefs

Cf. A000079, A002884, A171499.

Sequence in context: A024085 A080474 A079190 * A159865 A370680 A004806

Adjacent sequences: A203300 A203301 A203302 * A203304 A203305 A203306

Keyword

nonn

Author

Clark Kimberling, Jan 01 2012

Status

approved