A084148 Numerators of terms in the Pippenger product.

2, 8, 1152, 1605632, 43913893117952, 98583626709555431615548620800, 197241992148713072661201501950348880945923403897735704916000768

Offset

1,1

Links

G. C. Greubel, Table of n, a(n) for n = 1..10

Nicholas Pippenger, An infinite product for e, The American Mathematical Monthly, Vol. 87, No. 5 (1980), p. 391.

Eric Weisstein's World of Mathematics, Pippenger Product.

Formula

From Amiram Eldar, Apr 10 2022: (Start)

a(n) = numerator(((2^(n-1)-1)!!*(2^n)!!/((2^(n-1))!!*(2^n-1)!!))^2/2).

Product_{n>=1} (a(n)/A084149(n))^(1/2^n) = e/2 (A019739). (End)

a(n) = numerator( 2^(2^n -1)*((2^(n-1))!)^6 / (((2^n)!)^2 * ((2^(n-2))!)^4) ), with a(1) = 2. - G. C. Greubel, Oct 13 2022

Mathematica

a[n_] := Numerator[((2^(n - 1) - 1)!!*(2^n)!!/((2^(n - 1))!!*(2^n - 1)!!))^2/2]; Array[a, 7] (* Amiram Eldar, Apr 10 2022 *)

Prog

(Magma)
F:=Factorial;
A084148:= func< n | n eq 1 select 2 else Round(Numerator( 2^(2^n -1)*(F(2^(n-1)))^6 / ((F(2^n))^2 * (F(2^(n-2)))^4) )) >;
[A084148(n): n in [1..10]]; // G. C. Greubel, Oct 13 2022
(SageMath)
f=factorial
def A084148(n): return 2 if (n==1) else numerator( 2^(2^n -1)*(f(2^(n-1)))^6 / ((f(2^n))^2 * (f(2^(n-2)))^4) )
[A084148(n) for n in range(1, 10)] # G. C. Greubel, Oct 13 2022

Crossrefs

Cf. A019739, A084149 (denominators).

Sequence in context: A322142 A061591 A103085 * A014115 A014116 A027668

Adjacent sequences: A084145 A084146 A084147 * A084149 A084150 A084151

Keyword

frac,nonn

Author

Eric W. Weisstein, May 15 2003

Status

approved