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 A098118 a(n) = n!*[x^n](log(x+1)*sum(j=0..n, C(2*n,j)*x^j)). 11

%I

%S 1,7,74,1066,19524,434568,11393808,343976400,11752855200,448372820160,

%T 18892607771520,871406506494720,43669963405555200,2362804077652300800,

%U 137275789612950374400,8523776656311156172800,563309040416875548364800

%N a(n) = n!*[x^n](log(x+1)*sum(j=0..n, C(2*n,j)*x^j)).

%C Previous name was: Sum of all matrix elements of n X n Hilbert matrix M(i,j) = 1/(i+j-1) (i,j = 1..n) multiplied by (2*n-1)!/n!.

%C Let A(i, j) denote an infinite array such that the i-th row of this array is the sequence obtained by applying the partial sum operator i times to the harmonic sequence. For example, the first row starts as 1, 5/2, 13/3, ..., and the next row begins with 1, 7/2, 47/6, and so forth. Then a(n) equals n!*A(n, n) for all n. - _John M. Campbell_, Jan 20 2019

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HilbertMatrix.html">Hilbert Matrix.</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number.</a>

%F a(n) = (2*n-1)!/n!*Sum(Sum(1/(i+j-1), {i, 1, n}), {j, 1, n}).

%F a(n) = 2*(2*n-1)!/(n-1)!*H'(2n), where H'(2n) = H(2n) - H(n), H'(n) = Sum[(1/k)*(-1)^(k+1), (k, 1, n}] is an alternate signs Harmonic number, H(n) = Sum[1/k, (k, 1, n}] is a Harmonic number, H[n] = A001008/A002805. - _Alexander Adamchuk_, Oct 25 2004

%F Sum_{k>0} a(k) * k! * x^(2*k + 1) / (2*k + 1)! = F(-1) + F((1 - x)/2) + log(2) * log((1 + x) / (1 - x)) / 2 + log((1 + x) / 2) * log((1 - x) / 2) / 2 where F(x) = Li_2(x) is the dilogarithm function. - _Michael Somos_, Dec 09 2013

%F 2 * A078791(n) = a(n) * A000984(n). - _Michael Somos_, Apr 14 2015

%F a(n) = (2*n)!/n! * Sum_{k = 1..n} 1/(n + k). Column 1 of A257635. - _Peter Bala_, Nov 05 2015

%F E.g.f.: (log((sqrt(1-4*x)+1)/2)*(-3*x+sqrt(1-4*x)*(x-1)+1))/(4*x^2+sqrt(1-4*x)*(3*x-1)-5*x+1). - _Vladimir Kruchinin_, Jun 04 2016

%F a(n) = hypergeom([1,1,1-n], [2,n+2], 1)*n*(2*n)!/(n+1)!. - _Peter Luschny_, Jun 11 2016

%F a(n) ~ log(2) * 2^(2*n + 1/2) * n^n / exp(n). - _Vaclav Kotesovec_, Jul 10 2016

%e n=2: HilbertMatrix[n,n]

%e 1 1/2

%e 1/2 1/3

%e so a(2) = (2*2-1)! / 2! * (1 + 1/2 + 1/2 + 1/3) = 7.

%e The n X n Hilbert matrix begins:

%e 1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 ...

%e 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 ...

%e 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 ...

%e 1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 ...

%e 1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12 ...

%e 1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...

%e G.f. = x + 7*x^2 + 74*x^3 + 1066*x^4 + 19524*x^5 + 434568*x^6 + ...

%p A098118 := n -> n!*coeff(series(ln(x+1)*add(binomial(2*n,j)*x^j, j=0..n), x, n+1), x, n): seq(A098118(n),n=1..17); # _Peter Luschny_, Jan 18 2015

%p A098118 := n -> hypergeom([1,1,1-n],[2,n+2],1)*n*(2*n)!/(n+1)!:

%p seq(simplify(A098118(n)), n=1..17); # _Peter Luschny_, Jun 11 2016

%t Table[(2n - 1)!/n! Sum[ 1/(i + j - 1), {i, n}, {j, n}], {n, 17}]

%t a[ n_] := If[ n < 1, 0, (2 n)! / n! Sum[ -(-1)^k / k, {k, 2 n}]]; (* _Michael Somos_, Dec 09 2013 *)

%t a[ n_] := If[ n < 1, 0, (2 n - 1)! / n! Sum[ 1 / (i + j - 1), {i, n}, {j, n}]]; (* _Michael Somos_, Apr 14 2015 *)

%t a[ n_] := If[ n < 1, 0, n! SeriesCoefficient[ (Log[ EllipticNomeQ[ m] / (m/16)]) EllipticK[ m] 16^n / (Binomial[2 n, n] 2 Pi), {m, 0, n}]]; (* _Michael Somos_, Apr 14 2015 *)

%t a[ n_] := If[ n < 1, 0, (2 n + 1)! / n! SeriesCoefficient[ PolyLog[2, -1] + PolyLog[2, (1 - x)/2] + Log[(1 + x)/2] Log[(1 - x)/2]/2 + Log[(1 + x)/(1 - x)] Log[2]/2, {x, 0, 2 n + 1}]]; (* _Michael Somos_, Apr 14 2015 *)

%o (PARI) {a(n) = if( n<1, 0, (2*n)! / n! * sum( k=1, 2*n, -(-1)^k / k))}; /* _Michael Somos_, Dec 09 2013 */

%Y Cf. A000984, A001008, A002805, A005249, A078791, A086881, A257635 (column 1).

%K nonn,easy

%O 1,2

%A _Alexander Adamchuk_, Oct 25 2004

%E New name from _Peter Luschny_, Jan 19 2015

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Last modified August 19 08:08 EDT 2019. Contains 326115 sequences. (Running on oeis4.)