login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A097344 Numerators in binomial transform of 1/(n+1)^2. 4
1, 5, 29, 103, 887, 1517, 18239, 63253, 332839, 118127, 2331085, 4222975, 100309579, 184649263, 1710440723, 6372905521, 202804884977, 381240382217, 13667257415003, 25872280345103, 49119954154463, 93501887462903, 4103348710010689, 7846225754967739, 75162749477272151 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Is this identical to A097345? - Aaron Gulliver, Jul 19 2007. The answer turns out to be No - see A134652.
If the putative formula a(n)=A081528(n) sum{k=0..n, binomial(n, k)/(k+1)^2} were true, then this sequence coincides with A097345 according to Mathar's notes. However, the term n=9 in the binomial transform of 1/(n+1)^2 has the denominator 5040=A081528(9)/4=A081528(10)/5. So the formula cannot be true. - M. F. Hasler, Jan 25 2008
a(n) is also the numerator of u(n+1) with u(n) = (1/n)*Sum_{k=1..n} (2^k-1)/k and we have the formula: polylog(2,x/(1-x)) = Sum_{n>=1} u(n)*x^n on the interval [-1/2, 1/2]. - Groux Roland, Feb 01 2009
LINKS
FORMULA
a(n) = numerator(b(n)), b(n) = 1/((n+1)^2)*((n)*(3*n+1)*b(n-1)-2*(n-1)*(n)*b(n-2)+1). - Vladimir Kruchinin, May 31 2016
EXAMPLE
The first values of the binomial transform of 1/(n+1)^2 are 1, 5/4, 29/18, 103/48, 887/300, 1517/360, 18239/2940, 63253/6720, 332839/22680, 118127/5040, 2331085/60984, ...
MAPLE
f:=n->numer(add( binomial(n, k)/(k+1)^2, k=0..n));
MATHEMATICA
Table[HypergeometricPFQ[{1, 1, -n}, {2, 2}, -1] // Numerator, {n, 0, 24}] (* Jean-François Alcover, Oct 14 2013 *)
PROG
(PARI) A097344(n)=numerator(sum(k=0, n, binomial(n, k)/(k+1)^2)) \\ M. F. Hasler, Jan 25 2008
(Python)
from fractions import Fraction
A097344_list, tlist = [1], [Fraction(1, 1)]
for i in range(1, 100):
for j in range(len(tlist)):
tlist[j] *= Fraction(i, i-j)
tlist += [Fraction(1, (i+1)**2)]
A097344_list.append(sum(tlist).numerator) # Chai Wah Wu, Jun 04 2015
(Maxima)
a(n):=if n<0 then 1 else 1/((n+1)^2)*((n)*(3*n+1)*a(n-1)-2*(n-1)*(n)*a(n-2)+1);
makelist(num(a(n), n, 0, 10); /* Vladimir Kruchinin, Jun 01 2016 */
(Sage)
def A097344_list(size):
R, L = [1], [1]
inc = sqr = 1
for i in range(1, size):
for j in range(i):
L[j] *= i / (i - j)
inc += 2; sqr += inc
L.extend(1 / sqr)
R.append(sum(L).numerator())
return R
A097344_list(50) # (after Chai Wah Wu) Peter Luschny, Jun 05 2016
CROSSREFS
Sequence in context: A205172 A139856 A097345 * A153076 A034700 A057721
KEYWORD
easy,nonn,frac
AUTHOR
Paul Barry, Aug 06 2004
EXTENSIONS
Edited and corrected by Daniel Glasscock (glasscock(AT)rice.edu), Jan 04 2008 and M. F. Hasler, Jan 25 2008
Moved comment on numerators of a logarithmic g.f. over to A097345 - R. J. Mathar, Mar 04 2010
STATUS
approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)