A288094 - OEIS

%I #12 May 21 2023 17:11:45

%S 3,8,8,6,9,5,9,6,5,3,7,4,0,8,4,3,4,9,5,4,2,8,5,6,9,9,1,0,9,3,6,7,0,5,

%T 6,7,2,7,0,5,3,0,9,5,8,7,5,2,0,1,6,0,4,8,5,8,0,4,3,9,5,3,3,8,6,9,1,7,

%U 0,3,7,6,2,2,7,6,7,8,4,7,3,1,7,5,6,7,6,4,0,6,0,6,4,5,8,3,0,0,1,7,4,4,7,6

%N Decimal expansion of m(7) = Sum_{n>=0} 1/n!7, the 7th reciprocal multifactorial constant.

%H G. C. Greubel, <a href="/A288094/b288094.txt">Table of n, a(n) for n = 1..10000</a>

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

%F 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.

%e 3.88695965374084349542856991093670567270530958752016048580439533869...

%t 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[7], 10, 104][[1]]

%t RealDigits[Total[Table[1/Times@@Range[n,1,-7],{n,0,500}]],10,120][[1]] (* _Harvey P. Dale_, May 21 2023 *)

%o (PARI) default(realprecision, 105); (1/7)*exp(1/7)*(7 + sum(k=1,6, 7^(k/7)*(gamma(k/7) - incgam(k/7, 1/7)))) \\ _G. C. Greubel_, Mar 28 2019

%o (Magma) SetDefaultRealField(RealField(105)); (1/7)*Exp(1/7)*(7 + (&+[7^(k/7)*Gamma(k/7, 1/7): k in [1..6]])); // _G. C. Greubel_, Mar 28 2019

%o (Sage) numerical_approx((1/7)*exp(1/7)*(7 + sum(7^(k/7)*(gamma(k/7) - gamma_inc(k/7, 1/7)) for k in (1..6))), digits=105) # _G. C. Greubel_, Mar 28 2019

%Y Cf. A114799 (n!7), A143280 (m(2)), A288055 (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), this sequence (m(7)), A288095 (m(8)), A288096 (m(9)).

%K nonn,cons

%O 1,1

%A _Jean-François Alcover_, Jun 05 2017