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

1, 7, 74, 1066, 19524, 434568, 11393808, 343976400, 11752855200, 448372820160, 18892607771520, 871406506494720, 43669963405555200, 2362804077652300800, 137275789612950374400, 8523776656311156172800, 563309040416875548364800

Offset

1,2

Comments

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!.

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

Links

Seiichi Manyama, Table of n, a(n) for n = 1..366

Eric Weisstein's World of Mathematics, Hilbert Matrix.

Eric Weisstein's World of Mathematics, Harmonic Number.

Formula

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

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

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

2 * A078791(n) = a(n) * A000984(n). - Michael Somos, Apr 14 2015

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

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

a(n) = hypergeom([1,1,1-n], [2,n+2], 1)*n*(2*n)!/(n+1)!. - Peter Luschny, Jun 11 2016

a(n) ~ log(2) * 2^(2*n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, Jul 10 2016

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

Example

n=2: HilbertMatrix[n,n]
  1 1/2
  1/2 1/3
so a(2) = (2*2-1)! / 2! * (1 + 1/2 + 1/2 + 1/3) = 7.
The n X n Hilbert matrix begins:
  1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 ...
  1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 ...
  1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 ...
  1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 ...
  1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12 ...
  1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...
G.f. = x + 7*x^2 + 74*x^3 + 1066*x^4 + 19524*x^5 + 434568*x^6 + ...

Maple

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
A098118 := n -> hypergeom([1, 1, 1-n], [2, n+2], 1)*n*(2*n)!/(n+1)!:
seq(simplify(A098118(n)), n=1..17); # Peter Luschny, Jun 11 2016
A098118 := n -> sum(abs(Stirling1(n, k))*k*(n+1)^(k-1), k=1..n):
seq(A098118(n), n=1..17); # Ondrej Kutal, Oct 20 2021

Mathematica

Table[(2n - 1)!/n! Sum[ 1/(i + j - 1), {i, n}, {j, n}], {n, 17}]
a[ n_] := If[ n < 1, 0, (2 n)! / n! Sum[ -(-1)^k / k, {k, 2 n}]]; (* Michael Somos, Dec 09 2013 *)
a[ n_] := If[ n < 1, 0, (2 n - 1)! / n! Sum[ 1 / (i + j - 1), {i, n}, {j, n}]]; (* Michael Somos, Apr 14 2015 *)
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 *)
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 *)

Prog

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

Crossrefs

Cf. A000984, A001008, A002805, A005249, A078791, A008275, A086881.

Cf. A257635 (column 1).

Sequence in context: A365844 A341330 A266305 * A097821 A337387 A054745

Adjacent sequences: A098115 A098116 A098117 * A098119 A098120 A098121

Keyword

nonn,easy

Author

Alexander Adamchuk, Oct 25 2004

Extensions

New name from Peter Luschny, Jan 19 2015

Status

approved