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A244664 Decimal expansion of Sum_{n >= 1} H(n,2)/n^2 where H(n,2) = A007406(n)/A007407(n) is the n-th harmonic number of order 2.

%I A244664 #24 Aug 31 2025 10:27:58
%S A244664 1,8,9,4,0,6,5,6,5,8,9,9,4,4,9,1,8,3,5,1,5,3,0,0,6,4,6,8,9,4,7,0,4,3,
%T A244664 8,2,9,8,5,5,8,1,4,1,6,5,8,5,7,7,7,2,0,8,8,4,4,5,2,0,8,3,7,7,0,2,7,2,
%U A244664 1,1,0,7,8,3,2,7,1,9,5,4,8,1,4,7,4,5,9,1,8,6,2,8,9,7,9,7,4,8,5,5
%N A244664 Decimal expansion of Sum_{n >= 1} H(n,2)/n^2 where H(n,2) = A007406(n)/A007407(n) is the n-th harmonic number of order 2.
%H A244664 Vincenzo Librandi, <a href="/A244664/b244664.txt">Table of n, a(n) for n = 1..1000</a>
%H A244664 Philippe Flajolet, Bruno Salvy, <a href="http://algo.inria.fr/flajolet/Publications/FlSa98.pdf">Euler Sums and Contour Integral Representations</a>, Experimental Mathematics 7:1 (1998) page 23.
%F A244664 Equals 7*Pi^4/360 = (7/4)*A013662.
%F A244664 From _Peter Bala_, Jul 27 2025: (Start)
%F A244664 Series acceleration formula:
%F A244664 Let s(n) = Sum_{k = 1..n} H(k,2)/k^2 and S(n) = Sum_{k = 1..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*s(n+k). It appears that S(n) converges much more rapidly to 7*Pi^4/360 than s(n).
%F A244664 For example, s(50) = 1.8(61...) is only correct to 2 decimal digits, while S(50) = 1.89406565899449183515 30064689470(06...) is correct to 32 decimal digits. (End)
%e A244664 1.894065658994491835153006468947043829855814165857772088445208377027211...
%t A244664 RealDigits[7/4*Zeta[4], 10, 100] // First
%o A244664 (PARI) 7*zeta(4)/4 \\ _Michel Marcus_, Jul 04 2014
%Y A244664 Cf. A007406, A007407, A013662.
%K A244664 nonn,cons,changed
%O A244664 1,2
%A A244664 _Jean-François Alcover_, Jul 04 2014