A091660 Decimal expansion of -PolyGamma(2,(1+sqrt(3))/2)/2.

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

Offset

0,1

References

Bruce C. Berndt and Robert A. Rankin, Ramanujan: Letters and Commentary, Amer. Math. Soc., 1995, p. 57.

Links

Table of n, a(n) for n=0..101.

Bruce C. Berndt and Atul Dixit, Ramanujan's Beautiful Integrals, Hardy-Ramanujan Journal, Vol. 43 (2020), pp. 69-82. See Theorem 3.2, p. 72.

G. H. Hardy, Srinivasa Ramanujan, obituary notice, Proceedings of the London Mathematical Society, Vol. s2-19, No. 1 (1921), pp. xl-xlix. See p. xlix, eq. (2).

Oskar Perron, Über die Preeceschen Kettenbrüche, Sitz. Bayer. Akad. Wiss. München Math. Phys. Kl. (1953), pp. 21-56.

C. T. Preece, Theorems Stated by Ramanujan (X), J. London Math. Soc., Vol. s1-6, No. 1 (1931), pp. 22-32.

Eric Weisstein's World of Mathematics, Ramanujan Continued Fractions.

Formula

Equals 2*Integral_{x >= 0} x^2*csch(x)/e^(sqrt(3)*x) dx.

Equals 1/(1 + 1^3/(1 + 1^3/(3 + 2^3/(1 + 2^3/(5 + 3^3/(1 + 3^3/(7 + 4^3/(1 + 1/(9 + 4^3/(1 + ... )))))))))). - Amiram Eldar, Jan 01 2025

Example

0.52693911354126361921721873388403709196385090030646...

Mathematica

RealDigits[-PolyGamma[2, (1+Sqrt[3])/2]/2, 10, 100][[1]] (* Vaclav Kotesovec, Aug 15 2015 *)

Prog

(PARI)
polygamma(n, x) = if (n == 0, psi(x), (-1)^(n+1)*n!*zetahurwitz(n+1, x));
-polygamma(2, (1+sqrt(3))/2)/2 \\ Gheorghe Coserea, Sep 30 2018
(PARI)
2*intnum(x=0, [oo, 11/4], x^2/(sinh(x)*exp(sqrt(3)*x))) \\ Gheorghe Coserea, Sep 30 2018

Crossrefs

Cf. A091659.

Sequence in context: A344964 A321074 A318384 * A307029 A292580 A301914

Adjacent sequences: A091657 A091658 A091659 * A091661 A091662 A091663

Keyword

nonn,cons

Author

Eric W. Weisstein, Jan 26 2004

Status

approved