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

1,2

Comments

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.

Links

Table of n, a(n) for n=1..15.

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.

Formula

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)).

Example

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.

Crossrefs

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

Sequence in context: A124155 A218764 A217454 * A005390 A253245 A168166

Adjacent sequences: A101628 A101629 A101630 * A101632 A101633 A101634

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Dec 23 2004

Status

approved