A027637 a(n) = Product_{i=1..n} (4^i - 1).

1, 3, 45, 2835, 722925, 739552275, 3028466566125, 49615367752825875, 3251543125681443718125, 852369269595510700600441875, 893773106866112632882108339078125, 3748755223447856814435325652920396921875

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The q-analog of double factorials (A000165) evaluated at q=2. - Michael Somos, Sep 12 2014

3^n*5^(floor(n/2))|a(n) for n>=1. - G. C. Greubel, Nov 21 2015

Given probability p = 1/4^n that an outcome will occur at the n-th stage of an infinite process, then starting at n=1, 1-a(n)/A053763(n+1) is the probability that the outcome has occurred up to and including the n-th iteration. The limiting ratio is 1-A100221 ~ 0.3114625. - Bob Selcoe, Mar 01 2016

Links

• G. C. Greubel, Table of n, a(n) for n = 0..50

Crossrefs

Cf. A000165.

Sequences of the form q-Pochhammer(n, q, q): A005329 (q=2), A027871 (q=3), this sequence (q=4), A027872 (q=5), A027873 (q=6), A027875 (q=7), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027879 (q=11), A027880 (q=12).

Cf. A053763, A100221.

Programs

Magma

[1] cat [&*[4^k-1: k in [1..n]]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

Maple

A027637 := proc(n)
    mul( 4^i-1,i=1..n) ;
end proc:
seq(A027637(n),n=0..8) ;

Mathematica

A027637 = Table[Product[4^i - 1, {i, n}], {n, 0, 9}] (* Alonso del Arte, Nov 14 2011 *)
a[ n_] := If[ n < 0, 0, Product[ (q^(2 k) - 1) / (q - 1), {k, n}] /. q -> 2]; (* Michael Somos, Sep 12 2014 *)
Abs@QPochhammer[4, 4, Range[0, 10]] (* Vladimir Reshetnikov, Nov 20 2015 *)

PARI

a(n) = prod(i=1, n, 4^i-1); \\ Michel Marcus, Nov 21 2015

SageMath

from sage.combinat.q_analogues import q_pochhammer
def A027637(n): return (-1)^n*q_pochhammer(n, 4, 4)
[A027637(n) for n in (0..15)] # G. C. Greubel, Aug 04 2022

Formula

a(n) ~ c * 2^(n*(n+1)), where c = Product_{k>=1} (1-1/4^k) = A100221 = 0.688537537120339715456514357293508184675549819378... . - Vaclav Kotesovec, Nov 21 2015

a(n) = 4^(binomial(n+1,2))*(1/4;1/4){n} = (4; 4){n}, where (a;q){n} is the q-Pochhammer symbol. - _G. C. Greubel, Dec 24 2015

G.f.: Sum_{n>=0} 4^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 4^k*x). - Ilya Gutkovskiy, May 22 2017

Sum_{n>=0} (-1)^n/a(n) = A100221. - Amiram Eldar, May 07 2023