A397477 Decimal expansion of Sum_{k>=0} binomial(2*k, k)/(2^(3*k)*(2*k+1)^3).

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

Offset

1,6

Links

Table of n, a(n) for n=1..105.

John M. Campbell, Paul Levrie, and Amrik Singh Nimbran, A natural companion to Catalan's constant, Journal of Classical Analysis, Vol. 18, No. 2 (2021), pp. 117-135. See p. 121, eq. (19).

Steven R. Finch, Central Binomial Coefficients, 2007. See p. 5.

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2024. See section 1.22, p. 78.

Anthony Sofo and Amrik Singh Nimbran, Euler-like sums via powers of log, arctan and arctanh functions, Integral Transforms and Special Functions, Vol. 31, No. 12 (2020), pp. 966-981; ResearchGate link. See Corollary 2.12.

Formula

Equals 4F3([1/2, 1/2, 1/2, 1/2], [3/2, 3/2, 3/2], 1/2), where 4F3 is a generalized hypergeometric function.

Equals (Pi^3/192 + Im(polylog(3, 1 + i)))/sqrt(2), where i is the imaginary unit (Finch, 2020).

Equals -sqrt(2)*L/2 + 23*sqrt(2)*Pi^3/768 + 3*sqrt(2)*Pi*log(2)^2/64, where L = Im(polylog(3, (i+1)/2)) (A355022) (Campbell et al., 2021).

Equals (log(2)*(8*G + Pi*log(2))/8 - A247508)/(2*sqrt(2)), where G is Catalan's constant (A006752).

3 formulas from Sofo and Nimbran (2020):

Equals (23*Pi^3/384 + 3*Pi*log(2)^2/32 - Sum_{k>=1} sin(Pi*k/4)/(2^(k/2)*k^3))/sqrt(2).

Equals (G*log(2)/2 + 5*Pi^3/96 + Pi*log(2)^2/16 - Sum_{k>=1} h'(2*k-1)/(2*(2*k-1)^2))/sqrt(2), where G is Catalan's constant (A006752), and h'(n) = Sum_{k=1..n} (-1)^(k+1)/(2*k-1).

Equals (23*Pi^3/384 + 3*Pi*log(2)^2/32 - Sum_{k>=1} (-1)^(k + 1)/4^k*(2/(4*k - 3)^3 + 2/(4*k - 2)^3 + 1/(4*k - 1)^3))/sqrt(2).

Example

1.010154692526277592574269864638251385119930037754978...

Mathematica

RealDigits[(Pi^3/192 + Im[PolyLog[3, 1 + I]])/Sqrt[2], 10, 120][[1]]

Prog

(PARI) (Pi^3/192 + imag(polylog(3, 1 + I)))/sqrt(2)

Crossrefs

Sum_{k>=0} binomial(2*k, k)/(2^(3*k)*(2*k+1)^m): A002193 (m=0), A093954 (m=1), A263353 (m=2), this constant (m=3), A397478 (m=4).

Cf. A000984, A002162, A006752, A247508, A355022.

Sequence in context: A011502 A347185 A345231 * A335576 A270841 A386674

Adjacent sequences: A397474 A397475 A397476 * A397478 A397481 A397482

Keyword

nonn,cons

Author

Amiram Eldar, Jun 27 2026

Status

approved