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%I #6 Jun 14 2026 11:37:51
%S 4,9,2,7,3,8,1,0,2,2,4,5,3,4,1,8,3,2,2,3,3,6,7,1,0,3,9,0,2,1,7,3,6,7,
%T 4,1,9,5,7,3,5,5,6,9,8,6,3,6,7,1,0,7,9,3,7,4,2,3,9,1,4,6,2,1,0,1,7,8,
%U 8,2,5,8,6,0,3,6,9,7,5,5,5,2,6,1,9,4,0,4,9,5,9,5,4,9,7,2,5,8,4,4,5,3,7,7,5
%N Decimal expansion of Sum_{k>=0} binomial(2*k, k)^2 * (H(2*k) - H(k) - log(2))^2 / 16^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.
%H Cornel Ioan Vălean, <a href="https://doi.org/10.1007/978-3-031-21262-8">More (Almost) Impossible Integrals, Sums, and Series</a>, Springer Cham, 2023. See section 4.4, "A Great Time with a Special Binoharmonic Series", p. 396, eq.(4.15), section 5.4, p. 457, and section 6.4, pp. 489-490.
%H <a href="/index/Ha#harmonic">Index entries for sequences related to harmonic numbers</a>.
%F Equals 2*G - 7*zeta(3)/(2*Pi), where G is Catalan's constant (A006752).
%e 0.49273810224534183223367103902173674195735569863671...
%t RealDigits[2*Catalan - 7*Zeta[3]/(2*Pi), 10, 120][[1]]
%o (PARI) 2*Catalan - 7*zeta(3)/(2*Pi)
%Y Cf. A000984, A001008, A002805, A002894.
%Y Cf. A002117, A002162, A006752, A221209.
%K nonn,cons
%O 0,1
%A _Amiram Eldar_, Jun 14 2026