A309327 a(n) = Product_{k=1..n-1} (4^k + 1).

1, 1, 5, 85, 5525, 1419925, 1455423125, 5962868543125, 97701601079103125, 6403069829921181503125, 1678532740564688125136703125, 1760070825503098980191468752703125, 7382273863761775568111978346806480703125, 123854010565759745011512941023673583762640703125

Offset: 0

Author

Ilya Gutkovskiy, Jun 06 2020

Keywords

• nonn

Links

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

Crossrefs

Cf. A027637, A052539, A053763.

Sequences of the form Product_{j=1..n-1} (m^j + 1): A000012 (m=0), A011782 (m=1), A028362 (m=2), A290000 (m=3), this sequence (m=4).

Programs

Magma

[n lt 2 select 1 else (&*[4^j +1: j in [1..n-1]]): n in [0..15]]; // G. C. Greubel, Feb 21 2021

Mathematica

Table[Product[4^k + 1, {k, 1, n - 1}], {n, 0, 13}]
Join[{1}, Table[4^(Binomial[n,2])*QPochhammer[-1/4, 1/4, n-1], {n,15}]] (* G. C. Greubel, Feb 21 2021 *)

PARI

a(n) = prod(k=1, n-1, 4^k + 1); \\ Michel Marcus, Jun 06 2020

Sage

from sage.combinat.q_analogues import q_pochhammer
[1]+[4^(binomial(n,2))*q_pochhammer(n-1, -1/4, 1/4) for n in (1..15)] # G. C. Greubel, Feb 21 2021

Formula

G.f. A(x) satisfies: A(x) = 1 + x * A(4*x) / (1 - x).

G.f.: Sum_{k>=0} 2^(k*(k - 1)) * x^k / Product_{j=0..k-1} (1 - 4^j*x).

a(0) = 1; a(n) = Sum_{k=0..n-1} 4^k * a(k).

a(n) ~ c * 2^(n*(n - 1)), where c = Product_{k>=1} (1 + 1/4^k) = 1.355909673863479380345544...

a(n) = 4^(binomial(n+1,2))*(-1/4; 1/4){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Feb 21 2021