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Leading term of Stirling's approximation to n!, sqrt(2*Pi)*n^(n+(1/2))/e^n, rounded to nearest integer.
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%I #13 Jan 24 2024 09:43:41

%S 0,1,2,6,24,118,710,4980,39902,359537,3598696,39615625,475687486,

%T 6187239475,86661001741,1300430722199,20814114415223,353948328666101,

%U 6372804626194309,121112786592293963,2422786846761133394,50888617325509644403,1119751494628234263302

%N Leading term of Stirling's approximation to n!, sqrt(2*Pi)*n^(n+(1/2))/e^n, rounded to nearest integer.

%H G. C. Greubel, <a href="/A005394/b005394.txt">Table of n, a(n) for n = 0..449</a>

%p a:= n-> round(sqrt(2*Pi*n)*(n/exp(1))^n):

%p seq(a(n), n=0..23); # _Alois P. Heinz_, Jan 24 2024

%t Table[Round[Sqrt[2*Pi]*Exp[-n]*n^(n + 1/2)], {n, 0, 100}] (* _G. C. Greubel_, Aug 16 2018 *)

%o (Magma) R:= RealField(); [Round(Sqrt(2*Pi)*Exp(-n)*n^(n + 1/2)): n in [0..100]]; // _G. C. Greubel_, Aug 16 2018

%Y Cf. A000142, A090583.

%K nonn

%O 0,3

%A _N. J. A. Sloane_

%E Corrected and extended by _Hugo Pfoertner_, Jan 10 2004