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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

R. J. Mathar, Notes on an attempt to prove that A097344 and A097345 are identical

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-Fran├žois 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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Last modified December 14 21:25 EST 2017. Contains 296020 sequences.