A390243 Decimal expansion of Sum_{k>=0} (-1)^k/Catalan(k).

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

Offset

0,1

Links

Table of n, a(n) for n=0..104.

Thomas Koshy and Zhenguang Gao, Convergence of a Catalan series, The College Mathematics Journal, Vol. 43, No. 2 (2012), pp. 141-146; alternative link. See p. 145, eq. (5).

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} A033999(k)/A000108(k).

Equals (14 - 24*log(phi)/sqrt(5))/25, where phi is the golden ratio (A001622).

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

Example

0.35340370833727806133307204818370939579620607238066...

Mathematica

RealDigits[(14 - 24*Log[GoldenRatio]/Sqrt[5])/25, 10, 120][[1]]

Prog

(PARI) my(g = quadgen(5)); (14 - 24*log(g)/(2*g-1))/25

Crossrefs

Cf. A000108, A001622, A002390, A033999.

Sum_{k>=0} m^k/Catalan(k): A390245 (m = -3), A390244 (m = -2), this constant (m = -1), A268813 (m = 1), 5 + A197723 (m = 2), A390242 (m = 3).

Sequence in context: A077934 A077950 A077973 * A374624 A210606 A254327

Adjacent sequences: A390240 A390241 A390242 * A390244 A390245 A390246

Keyword

nonn,cons,easy

Author

Amiram Eldar, Oct 30 2025

Status

approved