A288091 Decimal expansion of m(4) = Sum_{n>=0} 1/n!!!!, the 4th reciprocal multifactorial constant.

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

Offset

1,1

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Eric Weisstein's MathWorld, Reciprocal Multifactorial Constant

Formula

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.

Example

3.485944977453557745218809044046404795092682320881969407647249998...

Mathematica

m[4] = (1/4)*E^(1/4)*(4 + Sqrt[2]*(Gamma[1/4] - Gamma[1/4, 1/4]) + 2*(Sqrt[Pi] - Gamma[1/2, 1/4]) + 2*Sqrt[2]*(Gamma[3/4] - Gamma[3/4, 1/4])); RealDigits[m[4], 10, 104][[1]]

Prog

(PARI) default(realprecision, 100); (1/4)*exp(1/4)*(4+sqrt(2)*(gamma(1/4) - incgam(1/4, 1/4))+2*(sqrt(Pi) -incgam(1/2, 1/4))+2*sqrt(2)*(gamma(3/4) - incgam(3/4, 1/4))) \\ G. C. Greubel, Mar 28 2019
(Magma) SetDefaultRealField(RealField(100)); (1/4)*Exp(1/4)*(4 + Sqrt(2)* Gamma(1/4, 1/4) + 2*Gamma(1/2, 1/4) + 2*Sqrt(2)*Gamma(3/4, 1/4)) // G. C. Greubel, Mar 28 2019
(Sage) numerical_approx((1/4)*exp(1/4)*(4 + sqrt(2)*(gamma(1/4) - gamma_inc(1/4, 1/4)) + 2*(sqrt(pi) - gamma_inc(1/2, 1/4)) + 2*sqrt(2)*(gamma(3/4) - gamma_inc(3/4, 1/4))), digits=100) # G. C. Greubel, Mar 28 2019

Crossrefs

Cf. A007662 (n!!!!), A143280 (m(2)), A288055 (m(3)), this sequence (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), A288096 (m(9)).

Sequence in context: A340533 A050274 A262951 * A378706 A057926 A078766

Adjacent sequences: A288088 A288089 A288090 * A288092 A288093 A288094

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 05 2017

Status

approved