Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
reviewed
approved
proposed
reviewed
editing
proposed
From Robert Israel, Jan 16 2018: (Start)
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
a(n) = 2^(n*(n-1)*(n-2)/6) * Product_{1<=k<=n-1} (2^k-1)^(n-k). (End)
a(n) = Product_{k=0..n-2} ( 2^(k+1)^2 * QPochhammer(2^(-k-1); 2; k+1) ). - G. C. Greubel, Aug 31 2023
# First program
# Second program
(* First program *)
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}]
(* Second program *)
Table[Product[2^(k+1) -2^j, {k, 0, n-2}, {j, 0, k}], {n, 15}] (* G. C. Greubel, Aug 31 2023 *)
(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
(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
approved
editing
editing
approved
a(n) ~ 1/A335011 * 2^(n*(n-1)*(2*n-1)/6) * QPochhammer(1/2)^n. - Vaclav Kotesovec, May 19 2020
approved
editing
reviewed
approved
proposed
reviewed
editing
proposed
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
proposed
editing