A203305 Vandermonde determinant of the first n terms of (1,3,7,15,31,...).
1, 2, 48, 64512, 20808990720, 6658450862270054400, 8590449816558320728896700416000, 180165778137909187135292776823951671626301440000, 246665746050863452218796304775365273357060390005370386894553088000000
Offset: 1
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 1..21
Programs
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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
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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}] (* A203305 *) Table[v[n+1]/v[n], {n,z}] (* A028365 *) %/2 (* A203307 *) (* Second program *) 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 *) -
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
Formula
a(n) = Product_{k=1..n-1} Product_{j=1..k} (2^(k+1) - 2^j).
From Vaclav Kotesovec, Feb 18 2021: (Start)
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).
a(n) ~ 2^(n*(n^2 - 1)/3) * QPochhammer(1/2)^n / A335011. (End)
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
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