A390244 Decimal expansion of Sum_{k>=0} (-2)^k/Catalan(k).

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

Index entries for transcendental numbers.

Formula

Equals Sum_{k>=0} A122803(k)/A000108(k).

Equals (1 - log(2+sqrt(3))/sqrt(3))/3.

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

Example

0.079884667899684550822968581706531391932790756350941...

Mathematica

RealDigits[(1 - Log[2 + Sqrt[3]]/Sqrt[3])/3, 10, 120, -1][[1]]

Prog

(PARI) (1 - log(2+sqrt(3))/sqrt(3))/3

Crossrefs

Cf. A000108, A065918, A122803.

Sum_{k>=0} m^k/Catalan(k): A390245 (m = -3), this constant (m = -2), A390243 (m = -1), A268813 (m = 1), 5 + A197723 (m = 2), A390242 (m = 3).

Sequence in context: A197516 A019862 A330865 * A300444 A363377 A371135

Adjacent sequences: A390241 A390242 A390243 * A390245 A390246 A390247

Keyword

nonn,cons,easy

Author

Amiram Eldar, Oct 30 2025

Status

approved