OFFSET
0,1
COMMENTS
From Peter Bala, Jan 28 2015: (Start)
As a sum of positive terms, the constant equals Sum_{k >= 1} k/(k!*(k+1)!). If we set S(n) = Sum_{k >= 0} k^n/(k!*(k+1)!) for n >= 0, so this constant is S(1), then S(n) is an integral linear combination of S(0) and S(1). For example S(7) = 16*S(0) + 11*S(1). Cf. A086880. S(0) is A096789.
The Pierce expansion of this constant begins [1, 3, 14, 15, 26, 40, 43, 71, 83, 8120, ...] giving the alternating series representation for this constant 1 - 1/3 + 1/(3*14) - 1/(3*14*15) + 1/(3*14*15*26) - .... (End)
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..2500
Eric Weisstein's World of Mathematics, Pierce Expansion.
FORMULA
Equals exp(-2) * Sum_{k>=0} binomial(2*k,k)/(k+1)!. - Amiram Eldar, Jun 12 2021
EXAMPLE
0.68894844769873820405495001581186710536...
MATHEMATICA
digits = 113; NSum[(-1)^(n+1)*1/Product[1+Floor[k/2], {k, 1, n}], {n, 1, Infinity}, NSumTerms -> digits, Method -> "AlternatingSigns", WorkingPrecision -> digits+5] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 21 2014 *)
RealDigits[BesselI[2, 2], 10, 113][[1]] (* Jean-François Alcover, Nov 19 2015, after Peter Bala *)
PROG
(PARI) suminf(n=1, (-1)^(n+1)*1./prod(i=1, n, 1+floor(i/2)))
(PARI) suminf(k=1, k/(k!*(k+1)!)) \\ Michel Marcus, Feb 03 2015
(PARI) besseli(2, 2) \\ Altug Alkan, Nov 19 2015
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Ralf Stephan, Sep 11 2013
STATUS
approved