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

%I A203303 #29 Sep 01 2023 02:24:13
%S A203303 1,1,6,1008,20321280,203199794380800,4096245678214226116608000,
%T A203303 671169825411994707343327912777482240000,
%U A203303 3589459026274030507466469204160461571257625328222208000000,2511229721141086754031154605327661795863172723306019839389105937236728217600000000
%N A203303 Vandermonde determinant of the first n terms of (1,2,4,8,16,...).
%C A203303 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.
%H A203303 Robert Israel, <a href="/A203303/b203303.txt">Table of n, a(n) for n = 1..22</a>
%H A203303 Daniel M. Kane, Carlo Sanna, and Jeffrey Shallit, <a href="https://arxiv.org/abs/1801.04483">Waring's theorem for binary powers</a>, arXiv:1801.04483 [math.NT], Jan 13 2018.
%F A203303 From _Robert Israel_, Jan 16 2018: (Start)
%F A203303 a(n) = Product_{0 <= i < j <= n-1} (2^j - 2^i).
%F A203303 a(n) = 2^(n*(n-1)*(n-2)/6) * Product_{1<=k<=n-1} (2^k-1)^(n-k). (End)
%F A203303 a(n) ~ 1/A335011 * 2^(n*(n-1)*(2*n-1)/6) * QPochhammer(1/2)^n. - _Vaclav Kotesovec_, May 19 2020
%F A203303 a(n) = Product_{k=0..n-2} ( 2^(k+1)^2 * QPochhammer(2^(-k-1); 2; k+1) ). - _G. C. Greubel_, Aug 31 2023
%p A203303 # First program
%p A203303 with(LinearAlgebra):
%p A203303 a:= n-> Determinant(VandermondeMatrix([2^i$i=0..n-1])):
%p A203303 seq(a(n), n=1..12);  # _Alois P. Heinz_, Jul 23 2017
%p A203303 # Second program
%p A203303 f:= n -> 2^(n*(n-1)*(n-2)/6)*mul((2^k-1)^(n-k),k=1..n-1):
%p A203303 seq(f(n),n=1..12); # _Robert Israel_, Jan 16 2018
%t A203303 (* First program *)
%t A203303 f[j_]:= 2^(j-1); z = 15;
%t A203303 v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
%t A203303 Table[v[n], {n,z}]                     (* A203303 *)
%t A203303 Table[v[n+1]/v[n], {n,z}]              (* A002884 *)
%t A203303 Table[v[n]*v[n+2]/(2*v[n+1]^2), {n,z}]  (* A171499 *)
%t A203303 Table[FactorInteger[v[n]], {n,z}]
%t A203303 (* Second program *)
%t A203303 Table[Product[2^(k+1) -2^j, {k,0,n-2}, {j,0,k}], {n,15}] (* _G. C. Greubel_, Aug 31 2023 *)
%o A203303 (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
%o A203303 (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
%Y A203303 Cf. A000079, A002884, A171499.
%K A203303 nonn
%O A203303 1,3
%A A203303 _Clark Kimberling_, Jan 01 2012