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Numerators of terms in the Pippenger product.
2

%I #19 Oct 13 2022 15:28:06

%S 2,8,1152,1605632,43913893117952,98583626709555431615548620800,

%T 197241992148713072661201501950348880945923403897735704916000768

%N Numerators of terms in the Pippenger product.

%H G. C. Greubel, <a href="/A084148/b084148.txt">Table of n, a(n) for n = 1..10</a>

%H Nicholas Pippenger, <a href="https://www.jstor.org/stable/2321215">An infinite product for e</a>, The American Mathematical Monthly, Vol. 87, No. 5 (1980), p. 391.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PippengerProduct.html">Pippenger Product</a>.

%F From _Amiram Eldar_, Apr 10 2022: (Start)

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

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

%F 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

%t 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 *)

%o (Magma)

%o F:=Factorial;

%o 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) )) >;

%o [A084148(n): n in [1..10]]; // _G. C. Greubel_, Oct 13 2022

%o (SageMath)

%o f=factorial

%o 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) )

%o [A084148(n) for n in range(1,10)] # _G. C. Greubel_, Oct 13 2022

%Y Cf. A019739, A084149 (denominators).

%K frac,nonn

%O 1,1

%A _Eric W. Weisstein_, May 15 2003