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Factorial expansion of Gamma(3/4) = Sum_{n>=1} a(n)/n! where Gamma is Euler's gamma function.
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%I #27 Sep 08 2022 08:45:05

%S 1,0,1,1,2,0,2,0,7,2,1,5,1,12,12,12,12,5,7,9,4,20,10,9,6,17,20,18,7,6,

%T 11,9,24,3,22,8,24,33,35,33,31,12,0,27,6,31,37,37,27,31,6,24,7,17,12,

%U 1,39,41,51,48,21,8,15,26,46,52,43,39,7,21,60,24,58,21,57,58,36,60,25,7

%N Factorial expansion of Gamma(3/4) = Sum_{n>=1} a(n)/n! where Gamma is Euler's gamma function.

%H G. C. Greubel, <a href="/A068463/b068463.txt">Table of n, a(n) for n = 1..1000</a>

%e Gamma(3/4) = 1 + 0/2! + 1/3! + 1/4! + 2/5! + 0/6! + 2/7! + ...

%t With[{b = Gamma[3/4]}, Table[If[n == 1, Floor[b], Floor[n!*b]-n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* _G. C. Greubel_, Nov 27 2018 *)

%o (PARI) A068463(N=90,c=gamma(precision(.75,logint(N!,10)+1)))=vector(N,n,if(n>1,c=c%1*n,c)\1) \\ - _M. F. Hasler_, Nov 26 2018

%o (PARI) default(realprecision, 250); b = gamma(3/4); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ _G. C. Greubel_, Nov 27 2018

%o (Magma) SetDefaultRealField(RealField(250)); [Floor(Gamma(3/4))] cat [Floor(Factorial(n)*Gamma(3/4)) - n*Floor(Factorial((n-1))*Gamma(3/4)) : n in [2..80]]; // _G. C. Greubel_, Nov 27 2018

%o (Sage)

%o def A068463(n):

%o if (n==1): return floor(gamma(3/4))

%o else: return expand(floor(factorial(n)*gamma(3/4)) - n*floor(factorial(n-1)*gamma(3/4)))

%o [A068463(n) for n in (1..80)] # _G. C. Greubel_, Nov 27 2018

%Y Cf. A075874, A068465 (decimal expansion), A068464 (Gamma(1/4)).

%K nonn

%O 1,5

%A _Benoit Cloitre_, Mar 10 2002

%E Name edited and keywords cons, easy removed by _M. F. Hasler_, Nov 26 2018