A101627 Numerator of partial sums of a certain series.

1, 39, 241, 34883, 14039, 1516871, 7601151, 875425657, 7887002813, 7095769757767, 14199583385459, 75087685321529, 75113436870869, 927229349730873529, 927436191807263569, 305182576081725442901, 23479178371879154033, 37713848011377144613, 37717984058802320713, 135759786815564675620247

Offset

1,2

Comments

The denominators are given in A101628.

Second member (m=3) of a family defined in A101028.

The limit s=lim(s(n),n->infinity) with the s(n) defined below equals 8*sum(zeta(2*k+1)/3^(2*k),k=1..infinity) with Euler's (or Riemann's) zeta function. This limit is 12*(log(3)-1) = 1.18334746...; see the Abramowitz-Stegun (given in A101028) reference p. 259, eq. 6.3.15 with z=1/3 together with p. 258.

Links

Table of n, a(n) for n=1..20.

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].

Wolfdieter Lang, Rationals s(n) and more.

Formula

a(n)=numerator(s(n)) with s(n)=24*sum(1/((3*k-1)*(3*k)*(3*k+1)), k=1..n).

Example

s(3)= 24*(1/(2*3*4)+ 1/(5*6*7) + 1/(8*9*10)) = 241/210, hence a(3)=241 and A101628(3)=210.

Mathematica

Numerator[Accumulate[Table[8/(9k^3-k), {k, 20}]]]

Prog

(PARI) a(n) = numerator(24*sum(k=1, n, 1/((3*k-1)*(3*k)*(3*k+1))));

Crossrefs

Cf. A101028 (m=2), A101629 (m=4), A101631 (m=5).

Cf. A101628 (denominators).

Sequence in context: A266104 A190538 A190606 * A229639 A070146 A372930

Adjacent sequences: A101624 A101625 A101626 * A101628 A101629 A101630

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Dec 23 2004

Extensions

More terms from Michel Marcus, Mar 01 2022

Status

approved