A390245 Decimal expansion of Sum_{k>=0} (-3)^k/Catalan(k) (negated).

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

Offset

0,2

Links

Table of n, a(n) for n=0..105.

Ulrich Abel, Reciprocal Catalan Sums, solution to problem 11765, American Mathematical Monthly, Vol. 123, No. 4 (2016), pp. 405-406.

Feng Qi and Bai-Ni Guo, Integral Representations of the Catalan Numbers and Their Applications, Mathematics, Vol. 5, No. 3 (2017), Article 40. See section 6.3, p. 19.

Li Yin and Feng Qi, Several series identities involving the Catalan numbers, Transactions of A. Razmadze Mathematical Institute, Vol. 172, No. 3, Part A (2018), pp. 466-474. See p. 467.

Formula

Equals Sum_{k>=0} A352779(k)/A000108(k).

Equals (10 - 36*log((5+sqrt(21))/2)/sqrt(21))/49.

Equals hypergeom([1, 2], [1/2], -3/4).

Example

-0.047112866550150200022813493583559824824068779604816...

Mathematica

RealDigits[(10 - 36*Log[(5+Sqrt[21])/2] / Sqrt[21]) / 49, 10, 120, -1][[1]]

Prog

(PARI) (10 - 36*log((5+sqrt(21))/2)/sqrt(21))/49

Crossrefs

Cf. A000108, A352779.

Sum_{k>=0} m^k/Catalan(k): this constant (m = -3), A390244 (m = -2), A390243 (m = -1), A268813 (m = 1), 5 + A197723 (m = 2), A390242 (m = 3).

Sequence in context: A147864 A388942 A195789 * A367608 A021959 A188735

Adjacent sequences: A390242 A390243 A390244 * A390246 A390247 A390248

Keyword

nonn,cons,easy

Author

Amiram Eldar, Oct 30 2025

Status

approved