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
Keywords
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.
Programs
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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
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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
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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 *) -
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
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
Comments