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A203305 Vandermonde determinant of the first n terms of (1,3,7,15,31,...).

%I A203305 #17 Aug 31 2023 02:56:44
%S A203305 1,2,48,64512,20808990720,6658450862270054400,
%T A203305 8590449816558320728896700416000,
%U A203305 180165778137909187135292776823951671626301440000,246665746050863452218796304775365273357060390005370386894553088000000
%N A203305 Vandermonde determinant of the first n terms of (1,3,7,15,31,...).
%C A203305 Each term divides its successor, as in A028365 and A203307.
%H A203305 G. C. Greubel, <a href="/A203305/b203305.txt">Table of n, a(n) for n = 1..21</a>
%F A203305 a(n) = Product_{k=1..n-1} Product_{j=1..k} (2^(k+1) - 2^j).
%F A203305 From _Vaclav Kotesovec_, Feb 18 2021: (Start)
%F A203305 a(n) = (-1)^n * (2^(n*(n+1)*(2*n+1)/6 - 1) / QPochhammer(2,2,n)) * Product_{k=2..n} QPochhammer(1/2^k, 2, k).
%F A203305 a(n) ~ 2^(n*(n^2 - 1)/3) * QPochhammer(1/2)^n / A335011. (End)
%F A203305 a(n) = Product_{k=2..n} ( 2^(k+1)^2 * QPochhammer(2^(-k-1), 2, k+1) )/ (2^(k+1) - 1). - _G. C. Greubel_, Aug 30 2023
%t A203305 (* First program *)
%t A203305 f[j_]:= 2^j - 1; z = 15;
%t A203305 v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
%t A203305 Table[v[n], {n,z}]         (* A203305 *)
%t A203305 Table[v[n+1]/v[n], {n,z}]  (* A028365 *)
%t A203305 %/2                         (* A203307 *)
%t A203305 (* Second program *)
%t A203305 Table[(-1)^n * 2^(n*(n+1)*(2*n+1)/6 - 1) / QPochhammer[2, 2, n] * Product[QPochhammer[1/2^k, 2, k], {k, 2, n}], {n, 10}] (* _Vaclav Kotesovec_, Feb 18 2021 *)
%o A203305 (Magma) [1] cat [(&*[(&*[2^(k+1) - 2^j: j in [1..k]]): k in [1..n-1]]): n in [2..15]]; // _G. C. Greubel_, Aug 30 2023
%o A203305 (SageMath) [product(product(2^k - 2^j for j in range(1,k)) for k in range(2,n+1)) for n in range(1,16)] # _G. C. Greubel_, Aug 30 2023
%Y A203305 Cf. A000225, A005329, A028365, A203307, A335011.
%K A203305 nonn
%O A203305 1,2
%A A203305 _Clark Kimberling_, Jan 01 2012