(MAGMAMagma) 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

OEIS Server: https://oeis.org/edit/global/2944

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G. C. Greubel, <a href="/A288096/b288096.txt">Table of n, a(n) for n = 1..10000</a>

m(k) = (1/k)*exp(1/k)*(k+ + Sum_{j=1..k-1} (gamma(j/k)-) - gamma(j/k, 1/k)) where gamma(x) is the Euler gamma function and gamma(a,x) the incomplete gamma function.

(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

(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

(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

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)).

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allocated for Jean-François Alcover

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

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, 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, 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

1,1

Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/ReciprocalMultifactorialConstant.html">Reciprocal Multifactorial Constant</a>

m(k) = (1/k)*exp(1/k)*(k+Sum_{j=1..k-1} (gamma(j/k)-gamma(j/k, 1/k)) where gamma(x) is the Euler gamma function and gamma(a,x) the incomplete gamma function.

4.08137552016889854407110514660961069462641007731860758843485175...

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]]

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)).

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nonn,cons

Jean-François Alcover, Jun 05 2017

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