A369675 a(n) = Product_{k=0..n} (4^k + 4^(n-k)).

2, 25, 2312, 1690000, 9773138432, 454542400000000, 167983232813812416512, 499835663627223040000000000, 11821129880009981801801971612516352, 2251076882713432721110048178176000000000000, 3407215210591493267547957182357614317126952945713152, 41525058946342607360045945411073338768005424742400000000000000

Offset

0,1

Links

Table of n, a(n) for n=0..11.

Formula

a(n) = Product_{k=0..n} (4^k + 4^(n-k)).

a(n) = 4^(n*(n+1)) * Product_{k=0..n} (1/4^k + 1/4^(n-k)).

a(n) = 4^(n*(n+1)/2) * Product_{k=0..n} (1 + 1/4^(n-2*k)).

From Vaclav Kotesovec, Feb 07 2024: (Start)

a(n) ~ c * 4^(3*n^2/4 + n), where

c = 2.276671433133289... = QPochhammer(-1, 1/16)^2/2 if n is even and

c = 2.284052876870834... = sqrt(2) * QPochhammer(-4, 1/16)^2 / 25 if n is odd. (End)

Example

a(0) = (1 + 1) = 2;
a(1) = (1 + 4)*(4 + 1) = 25;
a(2) = (1 + 4^2)*(4 + 4)*(4^2 + 1) = 2312;
a(3) = (1 + 4^3)*(4 + 4^2)*(4^2 + 4)*(4^3 + 1) = 1690000;
a(4) = (1 + 4^4)*(4 + 4^3)*(4^2 + 4^2)*(4^3 + 4)*(4^4 + 1) = 9773138432;
a(5) = (1 + 4^5)*(4 + 4^4)*(4^2 + 4^3)*(4^3 + 4^2)*(4^4 + 4)*(4^5 + 1) = 454542400000000;
...
RELATED SERIES.
Let F(x) be the g.f. of A369557, then
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...

Prog

(PARI) {a(n) = prod(k=0, n, 4^k + 4^(n-k))}
for(n=0, 15, print1(a(n), ", "))

Crossrefs

Cf. A369673, A369674, A369676, A369557.

Sequence in context: A203747 A278273 A210836 * A088816 A323589 A141415

Adjacent sequences: A369672 A369673 A369674 * A369676 A369677 A369678

Keyword

nonn

Author

Paul D. Hanna, Feb 06 2024

Status

approved