A390242 Decimal expansion of Sum_{k>=0} 3^k/Catalan(k).

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

Offset

2,1

Comments

In general, Sum_{k>=0} x^k/Catalan(k) = 2*(sqrt(4-x)*(8+x) + 12*sqrt(x)*arctan(sqrt(x/(4-x))))/(4-x)^(5/2) for -4 < x < 4 (see PlanetMath).

Links

Table of n, a(n) for n=2..106.

Kunle Adegoke, Robert Frontczak, and Taras Goy, On a family of infinite series with reciprocal Catalan numbers, Axioms, Vol. 11, No. 4 (2022), Article 165. See p. 2.

Tewodros Amdeberhan, Xiao Guan, Lin Jiu, Victor H. Moll, and Christophe Vignat, A series involving Catalan numbers: Proofs and demonstrations, Elemente der Mathematik, Vol. 71, No. 3 (2016), pp. 109-121.

David Beckwith, Problem 11765, American Mathematical Monthly, Vol. 121, No. 3 (2014), p. 267; Reciprocal Catalan Sums, solution to problem 11765 by Ulrich Abel, ibid., Vol. 123, No. 4 (2016), pp. 405-406.

PlanetMath, Generating function for the reciprocal Catalan numbers.

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 6.2, p. 18.

Seán M. Stewart, The inverse versine function and sums containing reciprocal central binomial coefficients and reciprocal Catalan numbers, International Journal of Mathematical Education in Science and Technology, Vol. 53, No. 7 (2022), pp. 1955-1966. See p. 1963, eq. (27c).

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} A000244(k)/A000108(k).

Equals 22 + 8*sqrt(3)*Pi = 22 + 8*A002194*A000796.

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

Example

65.53118474162122841425787818341173757481758995422577...

Mathematica

RealDigits[22 + 8*Sqrt[3]*Pi, 10, 120][[1]]

Prog

(PARI) 22 + 8*sqrt(3)*Pi

Crossrefs

Cf. A000244, A000108, A000796, A002194.

Similar constant: A145428 (Sum_{k>=0} 3^k/binomial(2*k,k)).

Sum_{k>=0} m^k/Catalan(k): A390245 (m = -3), A390244 (m = -2), A390243 (m = -1), A268813 (m = 1), 5 + A197723 (m = 2), this constant (m = 3).

Sequence in context: A351055 A359284 A157295 * A396211 A011485 A084339

Adjacent sequences: A390239 A390240 A390241 * A390243 A390244 A390245

Keyword

nonn,cons,easy

Author

Amiram Eldar, Oct 30 2025

Status

approved