A309327 - OEIS

%I #82 Feb 21 2021 04:09:43

%S 1,1,5,85,5525,1419925,1455423125,5962868543125,97701601079103125,

%T 6403069829921181503125,1678532740564688125136703125,

%U 1760070825503098980191468752703125,7382273863761775568111978346806480703125,123854010565759745011512941023673583762640703125

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

%H G. C. Greubel, <a href="/A309327/b309327.txt">Table of n, a(n) for n = 0..50</a>

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

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

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

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

%F 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

%t Table[Product[4^k + 1, {k, 1, n - 1}], {n, 0, 13}]

%t Join[{1}, Table[4^(Binomial[n,2])*QPochhammer[-1/4, 1/4, n-1], {n,15}]] (* _G. C. Greubel_, Feb 21 2021 *)

%o (PARI) a(n) = prod(k=1, n-1, 4^k + 1); \\ _Michel Marcus_, Jun 06 2020

%o (Sage)

%o from sage.combinat.q_analogues import q_pochhammer

%o [1]+[4^(binomial(n,2))*q_pochhammer(n-1, -1/4, 1/4) for n in (1..15)] # _G. C. Greubel_, Feb 21 2021

%o (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

%Y Cf. A027637, A052539, A053763.

%Y 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).

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jun 06 2020