A288055 Decimal expansion of m(3) = Sum_{n>=0} 1/n!!!, the 3rd reciprocal multifactorial constant.

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

Offset: 1

Author

Jean-François Alcover, Jun 05 2017

Keywords

• nonn
• cons

Examples

			3.298913538088419003401217808261469769077803695683207090885045129...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Crossrefs

Cf. A007661 (n!!!), A143280 (m(2)), this sequence (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), A288096 (m(9)).

Programs

Magma

SetDefaultRealField(RealField(100)); (Exp(1/3)/3)*(3 + 3^(1/3)*Gamma(1/3, 1/3) + 3^(2/3)*Gamma(2/3, 1/3)); // G. C. Greubel, Mar 27 2019

Mathematica

p = Pochhammer;
m[3] = 5/2 + Sum[1/((-3)^k*((2+3*k)*p[1/3 -k, k])) + 1/((-3)^k*((1+3*k) *p[2/3 -k, k])) +(-1)^(1-k)/(3^k*(k*p[1-k, -1 +k])), {k, 1, Infinity}]
(* or *)
m[3] = (Exp[1/3]/3) (3 + 3^(1/3) (Gamma[1/3] - Gamma[1/3, 1/3]) + 3^(2/3) (Gamma[2/3] - Gamma[2/3, 1/3]));
RealDigits[m[3], 10, 102][[1]]

PARI

default(realprecision, 100); (exp(1/3)/3)*(3 + 3^(1/3)*(gamma(1/3) - incgam(1/3, 1/3)) + 3^(2/3)*(gamma(2/3) - incgam(2/3, 1/3))) \\ G. C. Greubel, Mar 27 2019

Sage

numerical_approx((exp(1/3)/3)*(3 + 3^(1/3)*(gamma(1/3) - gamma_inc(1/3, 1/3)) + 3^(2/3)*(gamma(2/3) - gamma_inc(2/3, 1/3))), digits=100) # G. C. Greubel, Mar 27 2019

Formula

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.