A229837 Decimal expansion of Sum_{n>=1} 1/(n*n!).

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

Offset

1,2

Links

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

Stephen Crowley, Two New Zeta Constants, arXiv:1207.1126 [math.NT], 2012, page 17.

Wikipedia, Logarithmic integral function.

Formula

Sum_{n >= 1} 1/(n*n!) = Ei(1)-gamma where Ei is the exponential integral and gamma is Euler's constant.

Also pFq(1,1; 2,2; 1) where pFq is the generalized hypergeometric function.

Also li(e)-gamma, e being the Euler constant (A001113) and li the logarithmic integral function. - Stanislav Sykora, May 09 2015

Continued fraction expansion: Ei(1) - gamma = 1/(1 - 1^3/(5 - 2^3/(11 -...-(n-1)^3/(n^2+n-1) -...))). See A061572. - Peter Bala, Feb 01 2017

From Amiram Eldar, Aug 01 2020: (Start)

Equals Sum_{k>=1} H(k)*k/(k+1)!, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

Equals Integral_{x=0..1} (exp(x) - 1)/x dx.

Equals -Integral_{x=0..1} exp(x)*log(x) dx.

Equals -Integral_{x=1..e} log(log(x)) dx. (End)

Equals e * Sum_{k>=1} (-1)^(k+1)*H(k)/k!, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Jun 25 2021

Example

1.3179021514544038948600088442492318379749012457927839928404611969976461...

Maple

evalf(Ei(1)-gamma, 120); # Vaclav Kotesovec, May 10 2015

Mathematica

RealDigits[ ExpIntegralEi[1] - EulerGamma, 10, 100] // First

Prog

(PARI) -Euler-real(eint1(-1)) \\ Charles R Greathouse IV, Oct 01 2013

Crossrefs

Cf. A001008, A001113, A001620, A002805, A091725, A061572, A264806.

Sequence in context: A271059 A121370 A137908 * A019639 A306566 A329943

Adjacent sequences: A229834 A229835 A229836 * A229838 A229839 A229840

Keyword

nonn,cons

Author

Jean-François Alcover, Oct 01 2013

Status

approved