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A288092 Decimal expansion of m(5) = Sum_{n>=0} 1/n!5, the 5th reciprocal multifactorial constant.

%I A288092 #16 Apr 12 2025 05:18:32
%S A288092 3,6,4,0,2,2,4,4,6,7,7,3,3,8,0,9,7,3,4,1,7,6,9,3,7,2,3,6,9,6,3,5,6,9,
%T A288092 0,6,0,6,3,2,4,0,9,5,1,6,9,6,8,8,4,2,5,9,9,4,5,2,9,5,5,7,6,3,0,8,3,6,
%U A288092 6,6,5,7,3,1,3,2,8,1,4,8,5,2,5,9,0,0,6,4,4,4,1,3,9,8,6,9,1,0,1,3,5,7
%N A288092 Decimal expansion of m(5) = Sum_{n>=0} 1/n!5, the 5th reciprocal multifactorial constant.
%H A288092 G. C. Greubel, <a href="/A288092/b288092.txt">Table of n, a(n) for n = 1..10000</a>
%H A288092 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ReciprocalMultifactorialConstant.html">Reciprocal Multifactorial Constant</a>
%F A288092 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 A288092 3.640224467733809734176937236963569060632409516968842599452955763...
%t A288092 m[5] = (1/5)*E^(1/5)*(5 + 5^(1/5)*(Gamma[1/5] - Gamma[1/5, 1/5]) + 5^(2/5)*(Gamma[2/5] - Gamma[2/5, 1/5]) + 5^(3/5)*(Gamma[3/5] - Gamma[3/5, 1/5]) + 5^(4/5)*(Gamma[4/5] - Gamma[4/5, 1/5])); RealDigits[m[5], 10, 102][[1]]
%o A288092 (PARI) default(realprecision, 105); (1/5)*exp(1/5)*(5 + sum(k=1,4, 5^(k/5)*(gamma(k/5) - incgam(k/5, 1/5)))) \\ _G. C. Greubel_, Mar 28 2019
%o A288092 (Magma) SetDefaultRealField(RealField(105)); (1/5)*Exp(1/5)*(5 + (&+[5^(k/5)*Gamma(k/5, 1/5): k in [1..4]])); // _G. C. Greubel_, Mar 28 2019
%o A288092 (Sage) numerical_approx((1/5)*exp(1/5)*(5 + sum(5^(k/5)*(gamma(k/5) - gamma_inc(k/5, 1/5)) for k in (1..4))), digits=105) # _G. C. Greubel_, Mar 28 2019
%Y A288092 Cf. A085157 (n!5), A143280 (m(2)), A288055 (m(3)), A288091 (m(4)), this sequence (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), A288096 (m(9)).
%K A288092 nonn,cons
%O A288092 1,1
%A A288092 _Jean-François Alcover_, Jun 05 2017