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A288055 Decimal expansion of m(3) = Sum_{n>=0} 1/n!!!, the 3rd reciprocal multifactorial constant.

%I A288055 #28 Apr 12 2025 05:18:41
%S A288055 3,2,9,8,9,1,3,5,3,8,0,8,8,4,1,9,0,0,3,4,0,1,2,1,7,8,0,8,2,6,1,4,6,9,
%T A288055 7,6,9,0,7,7,8,0,3,6,9,5,6,8,3,2,0,7,0,9,0,8,8,5,0,4,5,1,2,9,0,4,3,8,
%U A288055 7,3,5,1,8,4,5,7,3,1,4,1,7,5,0,0,9,7,7,0,1,2,0,4,8,1,7,7,6,9,1,2,5,7
%N A288055 Decimal expansion of m(3) = Sum_{n>=0} 1/n!!!, the 3rd reciprocal multifactorial constant.
%H A288055 G. C. Greubel, <a href="/A288055/b288055.txt">Table of n, a(n) for n = 1..10000</a>
%H A288055 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ReciprocalMultifactorialConstant.html">Reciprocal Multifactorial Constant</a>
%F A288055 m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} k^(j/k)*(gamma(j/k) - gamma(j/k, 1/k))) where gamma(x) is the Euler gamma function and gamma(a,x) the incomplete gamma function.
%e A288055 3.298913538088419003401217808261469769077803695683207090885045129...
%t A288055 p = Pochhammer;
%t A288055 m[3] = 5/2 + Sum[1/((-3)^k*((2+3*k)*p[1/3 -k, k])) + 1/((-3)^k*((1+3*k) *p[2/3 -k, k])) +(-1)^(1-k)/(3^k*(k*p[1-k, -1 +k])), {k, 1, Infinity}]
%t A288055 (* or *)
%t A288055 m[3] = (Exp[1/3]/3) (3 + 3^(1/3) (Gamma[1/3] - Gamma[1/3, 1/3]) + 3^(2/3) (Gamma[2/3] - Gamma[2/3, 1/3]));
%t A288055 RealDigits[m[3], 10, 102][[1]]
%o A288055 (PARI) default(realprecision, 100); (exp(1/3)/3)*(3 + 3^(1/3)*(gamma(1/3) - incgam(1/3, 1/3)) + 3^(2/3)*(gamma(2/3) - incgam(2/3, 1/3))) \\ _G. C. Greubel_, Mar 27 2019
%o A288055 (Magma) SetDefaultRealField(RealField(100)); (Exp(1/3)/3)*(3 + 3^(1/3)*Gamma(1/3, 1/3) + 3^(2/3)*Gamma(2/3, 1/3)); // _G. C. Greubel_, Mar 27 2019
%o A288055 (Sage) numerical_approx((exp(1/3)/3)*(3 + 3^(1/3)*(gamma(1/3) - gamma_inc(1/3, 1/3)) + 3^(2/3)*(gamma(2/3) - gamma_inc(2/3, 1/3))), digits=100) # _G. C. Greubel_, Mar 27 2019
%Y A288055 Cf. A007661 (n!!!), A143280 (m(2)), this sequence (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), A288096 (m(9)).
%K A288055 nonn,cons
%O A288055 1,1
%A A288055 _Jean-François Alcover_, Jun 05 2017