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A288055 Decimal expansion of m(3) = Sum_{n>=0} 1/n!!!, the 3rd reciprocal multifactorial constant. 8
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 (list; constant; graph; refs; listen; history; text; internal format)
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 A010372

Adjacent sequences:  A288052 A288053 A288054 * A288056 A288057 A288058

KEYWORD

nonn,cons

AUTHOR

Jean-Fran├žois Alcover, Jun 05 2017

STATUS

approved

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Last modified November 17 00:08 EST 2019. Contains 329209 sequences. (Running on oeis4.)