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A288096 Decimal expansion of m(9) = Sum_{n>=0} 1/n!9, the 9th reciprocal multifactorial constant.

%I A288096 #17 Apr 12 2025 05:18:15
%S A288096 4,0,8,1,3,7,5,5,2,0,1,6,8,8,9,8,5,4,4,0,7,1,1,0,5,1,4,6,6,0,9,6,1,0,
%T A288096 6,9,4,6,2,6,4,1,0,0,7,7,3,1,8,6,0,7,5,8,8,4,3,4,8,5,1,7,5,1,6,7,4,9,
%U A288096 3,4,8,7,6,3,9,0,3,3,3,5,9,9,2,1,0,5,4,2,4,2,3,0,5,7,2,0,3,5,9,0,7,4
%N A288096 Decimal expansion of m(9) = Sum_{n>=0} 1/n!9, the 9th reciprocal multifactorial constant.
%H A288096 G. C. Greubel, <a href="/A288096/b288096.txt">Table of n, a(n) for n = 1..10000</a>
%H A288096 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ReciprocalMultifactorialConstant.html">Reciprocal Multifactorial Constant</a>
%F A288096 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 A288096 4.08137552016889854407110514660961069462641007731860758843485175...
%t A288096 m[k_] := (1/k) Exp[1/k] (k + Sum[k^(j/k) (Gamma[j/k] - Gamma[j/k, 1/k]), {j, 1, k - 1}]); RealDigits[m[9], 10, 102][[1]]
%o A288096 (PARI) default(realprecision, 105); (1/9)*exp(1/9)*(9 + sum(k=1,8, 9^(k/9)*(gamma(k/9) - incgam(k/9, 1/9)))) \\ _G. C. Greubel_, Mar 28 2019
%o A288096 (Magma) SetDefaultRealField(RealField(105)); (1/9)*Exp(1/9)*(9 + (&+[9^(k/9)*Gamma(k/9, 1/9): k in [1..8]])); // _G. C. Greubel_, Mar 28 2019
%o A288096 (Sage) numerical_approx((1/9)*exp(1/9)*(9 + sum(9^(k/9)*(gamma(k/9) - gamma_inc(k/9, 1/9)) for k in (1..8))), digits=105) # _G. C. Greubel_, Mar 28 2019
%Y A288096 Cf. A114806 (n!9), A143280 (m(2)), A288055 (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), this sequence (m(9)).
%K A288096 nonn,cons
%O A288096 1,1
%A A288096 _Jean-François Alcover_, Jun 05 2017