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

1, 2, 48, 64512, 20808990720, 6658450862270054400, 8590449816558320728896700416000, 180165778137909187135292776823951671626301440000, 246665746050863452218796304775365273357060390005370386894553088000000

Offset

1,2

Comments

Each term divides its successor, as in A028365 and A203307.

Links

G. C. Greubel, Table of n, a(n) for n = 1..21

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A000225, A005329, A028365, A203307, A335011.

Sequence in context: A098694 A137592 A203778 * A191954 A344665 A212170

Adjacent sequences: A203302 A203303 A203304 * A203306 A203307 A203308

Keyword

nonn

Author

Clark Kimberling, Jan 01 2012

Status

approved