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

1,1

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

3.298913538088419003401217808261469769077803695683207090885045129...

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]]

Prog

(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
(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
(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

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)).

Sequence in context: A276023 A268822 A201926 * A081233 A050676 A356185

Adjacent sequences: A288052 A288053 A288054 * A288056 A288057 A288058

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 05 2017

Status

approved