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Factorial expansion of Gamma(2/3) = Sum_{n>=1} a(n)/n!.
1

%I #9 May 24 2022 00:10:04

%S 1,0,2,0,2,2,6,6,0,3,1,11,7,6,6,14,1,8,12,15,8,17,8,1,13,15,3,4,10,16,

%T 25,1,25,22,6,3,19,17,8,10,25,37,29,17,35,19,24,25,30,31,4,7,51,49,14,

%U 51,45,54,0,26,34,41,56,57,16,15,63,4,51,42,13,35,12,15,66,22,13,43,14,78

%N Factorial expansion of Gamma(2/3) = Sum_{n>=1} a(n)/n!.

%H <a href="https://oeis.org/index/Fa#facbase">Index entries for factorial base representation</a>

%e Gamma(2/3) = 1 + 0/2! + 2/3! + 0/4! + 2/5! + 2/6! + 6/7! + 6/8! + ...

%t With[{b = Gamma[2/3]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]]

%o (PARI) default(realprecision, 250); b = gamma(2/3); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))

%o (Magma) SetDefaultRealField(RealField(250)); [Floor(Gamma(2/3))] cat [Floor(Factorial(n)*Gamma(2/3)) - n*Floor(Factorial((n-1))*Gamma(2/3)) : n in [2..80]];

%o (Sage)

%o b=gamma(2/3);

%o def a(n):

%o if (n==1): return floor(b)

%o else: return expand(floor(factorial(n)*b) -n*floor(factorial(n-1)*b))

%o [a(n) for n in (1..80)]

%Y Cf. A073006 (decimal expansion), A030652 (continued fraction).

%Y Cf. A068463 (Gamma(3/4)), A068464 (Gamma(1/4)), A322508 (Gamma(1/3)).

%K nonn

%O 1,3

%A _G. C. Greubel_, Dec 12 2018