A007407 a(n) = denominator of Sum_{k=1..n} 1/k^2.

1, 4, 36, 144, 3600, 3600, 176400, 705600, 6350400, 1270080, 153679680, 153679680, 25971865920, 25971865920, 129859329600, 519437318400, 150117385017600, 150117385017600, 54192375991353600, 10838475198270720, 221193371393280

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• easy
• frac
• nice

Comments

Denominators of the Eulerian numbers T(-2,k) for k = 0,1..., if T(n,k) is extended to negative n by the recurrence T(n,k) = (k+1)*T(n-1,k) + (n-k)*T(n-1,k-1) (indexed as in A173018). - Michael J. Collins, Oct 10 2024

Examples

			1/1^2 + 1/2^2 + 1/3^2 = 1/1 + 1/4 + 1/9 = 49/36, so a(3) = 36. - _Jon E. Schoenfield_, Dec 26 2014
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1152 (terms 1..200 from T. D. Noe)
• D. Y. Savio, E. A. Lamagna and S.-M. Liu, Summation of harmonic numbers, pp. 12-20 of E. Kaltofen and S. M. Watt, editors, Computers and Mathematics, Springer-Verlag, NY, 1989.

Crossrefs

Cf. A007406 (numerators), A000290, A035166.

Programs

Haskell

import Data.Ratio ((%), denominator)
a007407 n = a007407_list !! (n-1)
a007407_list = map denominator $
                   scanl1 (+) $ map (1 %) $ tail a000290_list
-- Reinhard Zumkeller, Jul 06 2012

Maple

ZL:=n->sum(1/i^2, i=2..n): a:=n->floor(denom(ZL(n))): seq(a(n), n=1..21); # Zerinvary Lajos, Mar 28 2007

Mathematica

s=0;lst={};Do[s+=n^2/n^4;AppendTo[lst,Denominator[s]],{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[Denominator[Pi^2/6 - Zeta[2, x]], {x, 1, 22}] (* Artur Jasinski, Mar 03 2010 *)
Denominator[Accumulate[1/Range[30]^2]] (* Harvey P. Dale, Nov 08 2012 *)

PARI

a(n)=denominator(sum(k=1,n,1/k^2)) \\ Charles R Greathouse IV, Nov 20 2012

Python

from fractions import Fraction
def A007407(n): return sum(Fraction(1,k**2) for k in range(1,n+1)).denominator # Chai Wah Wu, Apr 03 2021

Formula

a(n) = denominator of (Pi^2)/6 - zeta(2, x). - Artur Jasinski, Mar 03 2010

a(n) = A001044(n) / gcd(A001819(n), A001044(n)). - Daniel Suteu, Dec 25 2016