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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 (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 Chai Wah Wu, Table of n, a(n) for n = 0..500 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 Cf. A097345, A134652. Sequence in context: A205172 A139856 A097345 * A153076 A034700 A057721 Adjacent sequences:  A097341 A097342 A097343 * A097345 A097346 A097347 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

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