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A203303 revision #20

A203303
Vandermonde determinant of the first n terms of (1,2,4,8,16,...).
6
1, 1, 6, 1008, 20321280, 203199794380800, 4096245678214226116608000, 671169825411994707343327912777482240000, 3589459026274030507466469204160461571257625328222208000000, 2511229721141086754031154605327661795863172723306019839389105937236728217600000000
OFFSET
1,3
COMMENTS
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.
LINKS
Daniel M. Kane, Carlo Sanna, and Jeffrey Shallit, Waring's theorem for binary powers, arXiv:1801.04483 [math.NT], Jan 13 2018.
FORMULA
a(n) = Product_{0 <= i < j <= n-1} (2^j - 2^i) = 2^(n*(n-1)*(n-2)/6) * Product_{1<=k<=n-1} (2^k-1)^(n-k). - Robert Israel, Jan 16 2018
MAPLE
with(LinearAlgebra):
a:= n-> Determinant(VandermondeMatrix([2^i$i=0..n-1])):
seq(a(n), n=1..12); # Alois P. Heinz, Jul 23 2017
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
MATHEMATICA
f[j_] := 2^(j - 1); z = 15;
v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}] (* A203303 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}] (* A002884 *)
Table[v[n] v[n + 2]/(2*v[n + 1]^2), {n, 1, z - 1}] (* A171499 *)
Table[FactorInteger[v[n]], {n, 1, z - 1}]
CROSSREFS
Cf. A000079.
Sequence in context: A024085 A080474 A079190 * A159865 A370680 A004806
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jan 01 2012
STATUS
editing