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%I #35 Apr 11 2020 06:04:41
%S 1,5,29,103,887,1517,18239,63253,332839,118127,2331085,4222975,
%T 100309579,184649263,1710440723,6372905521,202804884977,381240382217,
%U 13667257415003,25872280345103,49119954154463,93501887462903,4103348710010689,7846225754967739,75162749477272151
%N Numerators in binomial transform of 1/(n+1)^2.
%C Is this identical to A097345? - Aaron Gulliver, Jul 19 2007. The answer turns out to be No - see A134652.
%C 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
%C 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
%H Chai Wah Wu, <a href="/A097344/b097344.txt">Table of n, a(n) for n = 0..500</a>
%H R. J. Mathar, <a href="/A097345/a097345.pdf">Notes on an attempt to prove that A097344 and A097345 are identical</a>
%F 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
%e 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, ...
%p f:=n->numer(add( binomial(n,k)/(k+1)^2, k=0..n));
%t Table[HypergeometricPFQ[{1, 1, -n}, {2, 2}, -1] // Numerator, {n, 0, 24}] (* _Jean-François Alcover_, Oct 14 2013 *)
%o (PARI) A097344(n)=numerator(sum(k=0,n,binomial(n,k)/(k+1)^2)) \\ _M. F. Hasler_, Jan 25 2008
%o (Python)
%o from fractions import Fraction
%o A097344_list, tlist = [1], [Fraction(1,1)]
%o for i in range(1,100):
%o for j in range(len(tlist)):
%o tlist[j] *= Fraction(i,i-j)
%o tlist += [Fraction(1,(i+1)**2)]
%o A097344_list.append(sum(tlist).numerator) # _Chai Wah Wu_, Jun 04 2015
%o (Maxima)
%o 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);
%o makelist(num(a(n),n,0,10); /* _Vladimir Kruchinin_, Jun 01 2016 */
%o (Sage)
%o def A097344_list(size):
%o R, L = [1], [1]
%o inc = sqr = 1
%o for i in range(1, size):
%o for j in range(i):
%o L[j] *= i / (i - j)
%o inc += 2; sqr += inc
%o L.extend(1 / sqr)
%o R.append(sum(L).numerator())
%o return R
%o A097344_list(50) # (after _Chai Wah Wu_) _Peter Luschny_, Jun 05 2016
%Y Cf. A097345, A134652.
%K easy,nonn,frac
%O 0,2
%A _Paul Barry_, Aug 06 2004
%E Edited and corrected by Daniel Glasscock (glasscock(AT)rice.edu), Jan 04 2008 and _M. F. Hasler_, Jan 25 2008
%E Moved comment on numerators of a logarithmic g.f. over to A097345 - _R. J. Mathar_, Mar 04 2010
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