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

%I A098118 #68 May 20 2025 08:53:21
%S A098118 1,7,74,1066,19524,434568,11393808,343976400,11752855200,448372820160,
%T A098118 18892607771520,871406506494720,43669963405555200,2362804077652300800,
%U A098118 137275789612950374400,8523776656311156172800,563309040416875548364800
%N A098118 a(n) = n!*[x^n] (log(x+1) * Sum_{j=0..n} C(2*n,j)*x^j).
%C A098118 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 A098118 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 A098118 Seiichi Manyama, <a href="/A098118/b098118.txt">Table of n, a(n) for n = 1..366</a>
%H A098118 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HilbertMatrix.html">Hilbert Matrix</a>.
%H A098118 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>.
%F A098118 a(n) = ((2*n-1)!/n!)*Sum_{i=1..n} Sum_{j=1..n} 1/(i+j-1).
%F A098118 a(n) = 2*(2*n-1)!/(n-1)!*H'(2*n), where H'(2*n) = H(2*n) - H(n), H'(n) = Sum_{k=1..n} (1/k)*(-1)^(k+1) is an alternate signs Harmonic number, H(n) = Sum_{k=1..n} 1/k is a Harmonic number, H(n) = A001008/A002805. - _Alexander Adamchuk_, Oct 25 2004
%F A098118 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 A098118 2 * A078791(n) = a(n) * A000984(n). - _Michael Somos_, Apr 14 2015
%F A098118 a(n) = (2*n)!/n! * Sum_{k = 1..n} 1/(n + k). Column 1 of A257635. - _Peter Bala_, Nov 05 2015
%F A098118 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 A098118 a(n) = hypergeom([1,1,1-n], [2,n+2], 1)*n*(2*n)!/(n+1)!. - _Peter Luschny_, Jun 11 2016
%F A098118 a(n) ~ log(2) * 2^(2*n + 1/2) * n^n / exp(n). - _Vaclav Kotesovec_, Jul 10 2016
%F A098118 a(n) = Sum_{k=1..n} |s(n,k)|*k*(n+1)^(k-1) where s(n,k) are Stirling numbers of the first kind (A008275). - _Ondrej Kutal_, Oct 20 2021
%F A098118 a(n) = n! * [x^n] (-log(1 - x)/(1 - x)^(n+1)). - _Seiichi Manyama_, May 20 2025
%e A098118 n=2: HilbertMatrix[n,n]
%e A098118   1 1/2
%e A098118   1/2 1/3
%e A098118 so a(2) = (2*2-1)! / 2! * (1 + 1/2 + 1/2 + 1/3) = 7.
%e A098118 The n X n Hilbert matrix begins:
%e A098118   1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 ...
%e A098118   1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 ...
%e A098118   1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 ...
%e A098118   1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 ...
%e A098118   1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12 ...
%e A098118   1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...
%e A098118 G.f. = x + 7*x^2 + 74*x^3 + 1066*x^4 + 19524*x^5 + 434568*x^6 + ...
%p A098118 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 A098118 := n -> hypergeom([1,1,1-n],[2,n+2],1)*n*(2*n)!/(n+1)!:
%p A098118 seq(simplify(A098118(n)), n=1..17); # _Peter Luschny_, Jun 11 2016
%p A098118 A098118 := n -> sum(abs(Stirling1(n,k))*k*(n+1)^(k-1), k=1..n):
%p A098118 seq(A098118(n), n=1..17); # _Ondrej Kutal_, Oct 20 2021
%t A098118 Table[(2n - 1)!/n! Sum[ 1/(i + j - 1), {i, n}, {j, n}], {n, 17}]
%t A098118 a[ n_] := If[ n < 1, 0, (2 n)! / n! Sum[ -(-1)^k / k, {k, 2 n}]]; (* _Michael Somos_, Dec 09 2013 *)
%t A098118 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 A098118 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 A098118 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 A098118 (PARI) {a(n) = if( n<1, 0, (2*n)! / n! * sum( k=1, 2*n, -(-1)^k / k))}; /* _Michael Somos_, Dec 09 2013 */
%Y A098118 Cf. A000984, A001008, A002805, A005249, A078791, A008275, A086881.
%Y A098118 Cf. A257635 (column 1).
%K A098118 nonn,easy
%O A098118 1,2
%A A098118 _Alexander Adamchuk_, Oct 25 2004
%E A098118 New name from _Peter Luschny_, Jan 19 2015