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Numerator of partial sums of a certain series.
5

%I #22 Dec 01 2024 20:15:02

%S 1,37,1069,20575,1346153,1214756107,20699705479,850029466379,

%T 19572345658457,137116980686111,411600123273343,1482039573988769177,

%U 456179332236626381,32398234503565880731,1199020509231104363863

%N Numerator of partial sums of a certain series.

%C The denominators are given in A101632.

%C Third member (m=5) of a family defined in A101028.

%C The limit s = lim_{n->oo} s(n) with the s(n) defined below equals 24*Sum_{k>=1} zeta(2*k+1)/5^(2*k) with Euler's (or Riemann's) zeta function. This limit is -24*(gamma + Psi(1/5) + 5/2 + Pi*cot(Pi/5)/2) = 1.1954056019...; see a comment in A101028 following from the Abramowitz-Stegun reference (given in A101028) p. 259, eq. 6.3.15 with z=1/5 together with p. 258.

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H W. Lang: <a href="/A101631/a101631.txt">Rationals s(n,5) and more.</a>

%F a(n) = numerator(s(n)) where s(n) = 120*Sum_{k=1..n} 1/((5*k-1)*(5*k)*(5*k+1)) = 24*Sum_{k=1..n} 1/((5*k-1)*k*(5*k+1)).

%e s(3) = 120*(1/(4*5*6) + 1/(9*10*11) + 1/(14*15*16)) = 1069/924, hence a(3)=1069 and A101632(3)=924.

%Y Cf. A101028, A101627, A101629, members 2, 3, 4, resp.

%K nonn,frac,easy

%O 1,2

%A _Wolfdieter Lang_, Dec 23 2004