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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
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
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