A378003 Decimal expansion of Pi*G - 7*zeta(3)/4, where G = A006752.

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

Offset

0,1

Comments

Srivastava and Miller (1990) expressed this constant by Kampé de Fériet's generalized hypergeometric function. - Amiram Eldar, Apr 21 2026

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7.2, p. 55.

Links

Table of n, a(n) for n=0..99.

Ming-Po Chen and H. M. Srivastava, Some families of series representations for the Riemann zeta(3), Results in Mathematics, Vol. 33, No. 3 (1998), pp. 179-197. See p. 190, eq. (3.12).

Marcin Mazur and Bogdan V. Petrenko, On the conjectures of Atiyah and Sutcliffe, Geometriae Dedicata, Vol. 158, No. 1 (2012), pp. 329-342; arXiv preprint, arXiv:1102.4662 [math.AG], 2011. See p. 333, eq. (7).

Mohd Idris Qureshi, Junesang Choi, and Mohd Shaid Baboo, Certain identities involving the general Kampé de Fériet function and Srivastava's general triple hypergeometric series, Symmetry, Vol. 14, No. 12 (2022), Article 2502.

Hari M. Srivastava and Elizabeth A. Miller, A simple reducible case of double hypergeometric series involving Catalan's constant and Riemann's zeta‐function, International Journal of Mathematical Education in Science and Technology, Vol. 21, No. 3 (1990), pp. 375-377.

Hari M. Srivastava, Bhawna Gupta, Mohammad Idris Qureshi, and Mohd Shaid Baboo, Some summation theorems and transformations for hypergeometric functions of Kampé de Fériet and Srivastava, Georgian Mathematical Journal, Vol. 31, No. 5 (2024), pp. 885-897; ResearchGate link. See p. 4, eq. (1.11).

Eric Weisstein's World of Mathematics, Kampé de Fériet Function.

Formula

Equals Sum_{n>=1} (-1)^(n+1)/n^2 Sum_{k=0..n-1} 1/(2*k + 1) (see Finch).

From Amiram Eldar, Apr 21 2026: (Start)

Equals Sum_{k>=0} 2^(4*k-1) / ((k+1) * (2*k + 1)^2 * binomial(2*k, k)^2) (Finch, 2003).

Equals (Pi^2/4) * Sum_{k>=0} (2^(2*k-1) - 1) * zeta(2*k)/((k+1)*2^(4*k)) (Chen and Srivastava, 1998).

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

Formulas from Srivastava and Miller (1990):

Equals 2 * Integral_{x=0..1} arctan(x)^2/x dx.

Equals (1/2) * Integral_{x=0..Pi/2} x^2/sin(x) dx.

Equals -Integral_{x=0..Pi/2} x * log(tan(x/2)) dx. (End)

Example

0.773991201078871152328038383876510316276128388...

Mathematica

RealDigits[Pi Catalan-7Zeta[3]/4, 10, 100][[1]]

Prog

(PARI) Pi*Catalan-7*zeta(3)/4 \\ Charles R Greathouse IV, Feb 12 2025

Crossrefs

Cf. A000796, A002117, A006752, A378021.

Sequence in context: A220052 A153204 A199508 * A136478 A097903 A396004

Adjacent sequences: A378000 A378001 A378002 * A378004 A378005 A378006

Keyword

nonn,cons

Author

Stefano Spezia, Nov 14 2024

Status

approved