|
%I #14 Feb 07 2024 07:48:13
%S 2,25,2312,1690000,9773138432,454542400000000,167983232813812416512,
%T 499835663627223040000000000,11821129880009981801801971612516352,
%U 2251076882713432721110048178176000000000000,3407215210591493267547957182357614317126952945713152,41525058946342607360045945411073338768005424742400000000000000
%N a(n) = Product_{k=0..n} (4^k + 4^(n-k)).
%F a(n) = Product_{k=0..n} (4^k + 4^(n-k)).
%F a(n) = 4^(n*(n+1)) * Product_{k=0..n} (1/4^k + 1/4^(n-k)).
%F a(n) = 4^(n*(n+1)/2) * Product_{k=0..n} (1 + 1/4^(n-2*k)).
%F From _Vaclav Kotesovec_, Feb 07 2024: (Start)
%F a(n) ~ c * 4^(3*n^2/4 + n), where
%F c = 2.276671433133289... = QPochhammer(-1, 1/16)^2/2 if n is even and
%F c = 2.284052876870834... = sqrt(2) * QPochhammer(-4, 1/16)^2 / 25 if n is odd. (End)
%e a(0) = (1 + 1) = 2;
%e a(1) = (1 + 4)*(4 + 1) = 25;
%e a(2) = (1 + 4^2)*(4 + 4)*(4^2 + 1) = 2312;
%e a(3) = (1 + 4^3)*(4 + 4^2)*(4^2 + 4)*(4^3 + 1) = 1690000;
%e a(4) = (1 + 4^4)*(4 + 4^3)*(4^2 + 4^2)*(4^3 + 4)*(4^4 + 1) = 9773138432;
%e a(5) = (1 + 4^5)*(4 + 4^4)*(4^2 + 4^3)*(4^3 + 4^2)*(4^4 + 4)*(4^5 + 1) = 454542400000000;
%e ...
%e RELATED SERIES.
%e Let F(x) be the g.f. of A369557, then
%e F(1/4) = 2 + 25/4^2 + 2312/4^6 + 1690000/4^12 + 9773138432/4^20 + 454542400000000/4^30 + ... + a(n)/4^(n*(n+1)) + ... = 4.236976626306045459467696438142250301516563681...
%o (PARI) {a(n) = prod(k=0, n, 4^k + 4^(n-k))}
%o for(n=0, 15, print1(a(n), ", "))
%Y Cf. A369673, A369674, A369676, A369557.
%K nonn
%O 0,1
%A _Paul D. Hanna_, Feb 06 2024
| 