A101632 -id:A101632 - cp's OEIS frontend

A101631 Numerator of partial sums of a certain series.

1, 37, 1069, 20575, 1346153, 1214756107, 20699705479, 850029466379, 19572345658457, 137116980686111, 411600123273343, 1482039573988769177, 456179332236626381, 32398234503565880731, 1199020509231104363863

Offset: 1

Wolfdieter Lang, Dec 23 2004

The denominators are given in A101632.
Third member (m=5) of a family defined in A101028.
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.

			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.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
W. Lang: Rationals s(n,5) and more.

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

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)).

cp's OEIS Frontend

A101631 Numerator of partial sums of a certain series.

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