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A342033 Decimal expansion of m(10) = Sum_{n>=0} 1/n!10, the 10th reciprocal multifactorial constant.

%I A342033 #31 Feb 16 2025 08:34:01
%S A342033 4,1,6,5,2,4,3,7,6,5,5,5,8,3,8,4,5,9,0,7,8,7,2,6,2,4,1,0,4,4,5,5,6,0,
%T A342033 7,3,8,2,2,8,0,3,0,7,9,5,3,7,0,7,7,2,7,7,6,7,9,4,4,2,1,9,1,1,5,0,7,0,
%U A342033 5,8,4,7,7,3,0,9,8,7,2,5,6,8,6,2,3,2,0,1,2,7,4,8,4,2,8,6,9,3,3,8,4,1,3,8
%N A342033 Decimal expansion of m(10) = Sum_{n>=0} 1/n!10, the 10th reciprocal multifactorial constant.
%C A342033 m(k) can be proved to approach a harmonic series (and diverge) as k approaches infinity.
%H A342033 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ReciprocalMultifactorialConstant.html">Reciprocal Multifactorial Constant</a>
%F A342033 m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} (k^(j/k)*Gamma(j/k, 1/k))) where Gamma(a,x) the incomplete Gamma function.
%e A342033 4.165243765558384590787262...
%e A342033 For n=10, the series is equal to 1+summation from n=1 to 10 (1/n)=9901/2520.
%t A342033 Multifactorial[n_, k_] := Abs[Apply[Times, Range[-n, -1, k]]]
%t A342033 N[Sum[1/Multifactorial[n, 10], {n, 0, 10000}], 105]
%t A342033 (* or *)
%t A342033 ReciprocalFactorialSumConstant[k_] :=
%t A342033 1/k Exp[1/k] (k + Sum[k^(j/k) Gamma[j/k, 0, 1/k], {j, k - 1}])
%t A342033 N[ReciprocalFactorialSumConstant[10], 105]
%Y A342033 Cf. A143280 (m(2)), A288055 (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), A288096 (m(9)).
%K A342033 nonn,cons
%O A342033 1,1
%A A342033 _Bhoris Dhanjal_, Feb 26 2021