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A288091 Decimal expansion of m(4) = Sum_{n>=0} 1/n!!!!, the 4th reciprocal multifactorial constant.

%I A288091 #19 Apr 12 2025 05:18:37
%S A288091 3,4,8,5,9,4,4,9,7,7,4,5,3,5,5,7,7,4,5,2,1,8,8,0,9,0,4,4,0,4,6,4,0,4,
%T A288091 7,9,5,0,9,2,6,8,2,3,2,0,8,8,1,9,6,9,4,0,7,6,4,7,2,4,9,9,9,8,1,3,1,6,
%U A288091 1,3,1,7,2,2,9,0,0,5,6,6,2,9,6,4,0,2,2,1,4,4,6,9,7,5,9,8,6,0,1,8,6,8,5,9
%N A288091 Decimal expansion of m(4) = Sum_{n>=0} 1/n!!!!, the 4th reciprocal multifactorial constant.
%H A288091 G. C. Greubel, <a href="/A288091/b288091.txt">Table of n, a(n) for n = 1..10000</a>
%H A288091 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ReciprocalMultifactorialConstant.html">Reciprocal Multifactorial Constant</a>
%F A288091 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 A288091 3.485944977453557745218809044046404795092682320881969407647249998...
%t A288091 m[4] = (1/4)*E^(1/4)*(4 + Sqrt[2]*(Gamma[1/4] - Gamma[1/4, 1/4]) + 2*(Sqrt[Pi] - Gamma[1/2, 1/4]) + 2*Sqrt[2]*(Gamma[3/4] - Gamma[3/4, 1/4])); RealDigits[m[4], 10, 104][[1]]
%o A288091 (PARI) default(realprecision, 100); (1/4)*exp(1/4)*(4+sqrt(2)*(gamma(1/4) - incgam(1/4, 1/4))+2*(sqrt(Pi) -incgam(1/2, 1/4))+2*sqrt(2)*(gamma(3/4) - incgam(3/4, 1/4))) \\ _G. C. Greubel_, Mar 28 2019
%o A288091 (Magma) SetDefaultRealField(RealField(100)); (1/4)*Exp(1/4)*(4 + Sqrt(2)* Gamma(1/4, 1/4) + 2*Gamma(1/2, 1/4) + 2*Sqrt(2)*Gamma(3/4, 1/4)) // _G. C. Greubel_, Mar 28 2019
%o A288091 (Sage) numerical_approx((1/4)*exp(1/4)*(4 + sqrt(2)*(gamma(1/4) - gamma_inc(1/4, 1/4)) + 2*(sqrt(pi) - gamma_inc(1/2, 1/4)) + 2*sqrt(2)*(gamma(3/4) - gamma_inc(3/4, 1/4))), digits=100) # _G. C. Greubel_, Mar 28 2019
%Y A288091 Cf. A007662 (n!!!!), A143280 (m(2)), A288055 (m(3)), this sequence (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), A288096 (m(9)).
%K A288091 nonn,cons
%O A288091 1,1
%A A288091 _Jean-François Alcover_, Jun 05 2017