A241215 - OEIS

%I #12 Jan 20 2025 12:50:59

%S 1,8,0,1,6,1,3,2,6,8,0,4,3,4,1,2,9,0,3,7,2,9,4,8,8,9,4,2,0,2,0,8,8,8,

%T 4,3,0,3,1,3,7,7,5,8,2,7,7,8,7,8,9,3,3,0,0,8,7,3,3,9,4,9,2,5,4,8,0,4,

%U 4,4,8,1,8,8,4,0,8,9,3,3,3,7,5,3,0,9,4,5,7,4,3,3,0,4,2,7,1,9,3,1

%N Decimal expansion of Sum_{n>=1} H(n)^4/(n+1)^3 where H(n) is the n-th harmonic number.

%H Philippe Flajolet and 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 16.

%F Equals (37/2)*zeta(3)*zeta(4) - 5*zeta(2)*zeta(5) - (109/8)*zeta(7).

%F Equals (37/180)*Pi^4*zeta(3) - (5/6)*Pi^2*zeta(5) - (109/8)*zeta(7).

%e 1.80161326804341290372948894202088843...

%t 37/180*Pi^4*Zeta[3] - 5/6*Pi^2*Zeta[5] - 109/8*Zeta[7] // RealDigits[#, 10, 100]& // First

%o (PARI) 37/2*zeta(3)*zeta(4) - 5*zeta(2)*zeta(5) - 109/8*zeta(7) \\ _Stefano Spezia_, Jan 19 2025

%Y Cf. A016627, A083680, A102886, A152648, A152649, A152651, A233033, A233090, A238166, A238167, A239168, A238169, A238181, A238182, A238183, A240264.

%K nonn,cons

%O 1,2

%A _Jean-François Alcover_, Apr 17 2014