The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A195505 Numerator of Sum_{k=1..n} H(k)/k^2, where H(k) is the k-th harmonic number. 3
 1, 11, 341, 2953, 388853, 403553, 142339079, 1163882707, 31983746689, 32452469713, 43725835522403, 44184852180503, 97954699428176291, 98731028315167091, 99421162547987123, 800313205356878959, 3953829021224881128767, 3973669953994085875967 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Lim_{n-> infinity} (a(n)/A195506(n)) = 2*Zeta(3) [L. Euler]. Sum_{k = 1..n} H(k)/k^2 is an example of a multiple harmonic (star) sum. Euler's result Sum_{k = 1..inf} H(k)/k^2 = 2*zeta(3) was the first evaluation of a multiple zeta star value. - Peter Bala, Jan 31 2019 LINKS Seiichi Manyama, Table of n, a(n) for n = 1..768 Leonhard Euler, Meditationes circa singulare serierum genus, Novi. Comm. Acad. Sci. Petropolitanae, 20 (1775), 140-186. FORMULA From Peter Bala, Jan 31 2019: (Start) Let S(n) = Sum_{k = 1..n} H(k)/k^2. Then S(n) = 1 + (1 + 1/2^3)*(n-1)/(n+1) + (1/2^3 + 1/3^3)*(n-1)*(n-2)/((n+1)*(n+2)) + (1/3^3 + 1/4^3)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + ... S(n)/n  = 1 + (1/2^4 - 1)*(n-1)/(n+1) + (1/3^4 - 1/2^4)*(n-1)*(n-2)/((n+1)*(n+2)) + (1/4^4 - 1/3^4)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + ... For odd n >= 3, 1/2*S((n-1)/2) = (n-1)/(n+1) + 1/2^3*(n-1)*(n-3)/((n+1)*(n+3)) + 1/3^3*(n-1)*(n-3)*(n-5)/((n+1)*(n+3)*(n+5)) + .... Cf. A001008. See the Bala link in A036970. (End) EXAMPLE a(2) = 11 because 1 + (1 + 1/2)/2^2 = 11/8. The first few fractions are 1, 11/8, 341/216, 2953/1728, 388853/216000, 403553/216000, 142339079/74088000, 1163882707/592704000, ... = A195505/A195506. - Petros Hadjicostas, May 06 2020 MATHEMATICA s = 0; Table[s = s + HarmonicNumber[n]/n^2; Numerator[s], {n, 20}] (* T. D. Noe, Sep 20 2011 *) PROG (PARI) H(n) = sum(k=1, n, 1/k); a(n) = numerator(sum(k=1, n, H(k)/k^2)); \\ Michel Marcus, May 07 2020 CROSSREFS Cf. A001008, A002117, A036970, A195506 (denominators). Sequence in context: A215015 A219072 A280669 * A092609 A091537 A327943 Adjacent sequences:  A195502 A195503 A195504 * A195506 A195507 A195508 KEYWORD nonn,frac,easy AUTHOR Franz Vrabec, Sep 19 2011 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 14 12:27 EDT 2020. Contains 336480 sequences. (Running on oeis4.)