A058607 - OEIS

%I #41 Oct 26 2019 06:53:59

%S 1,9,110,1750,34524,814968,22424688,705173040,24956062560,

%T 981852505920,42517741069440,2009786716304640,102980287835712000,

%U 5685838994441088000,336540101841974016000,21258495023757610752000,1427473447879197261312000,101537097118783918986240000,7626891980577579870504960000

%N a(n) = (1 + 1/2 + 1/3 + ... + 1/n)*(2n-1)!/(n-1)!.

%H Seiichi Manyama, <a href="/A058607/b058607.txt">Table of n, a(n) for n = 1..365</a>

%H Hongwei Chen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Chen/chen21.html">Interesting Series Associated with Central Binomial Coefficients, Catalan Numbers and Harmonic Numbers</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.5, p. 3.

%F E.g.f.: log((1 + 1/sqrt(1-4*x))/2)/sqrt(1-4*x).

%F a(n) = n!*Sum_{k=1..n} (binomial(2*n,n-k)/k). - _Vladimir Kruchinin_, Mar 31 2016

%F a(n) = 2*(2*n-1)*a(n-1)+binomial(2*n-1,n)*(n-1)!, a(1)=1. - _Vladimir Kruchinin_, Jun 11 2016

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

%p A058607 := n -> hypergeom([1,1,1-n],[2,n+2],-1)*n*(2*n)!/(n+1)!:

%p seq(simplify(A058607(n)),n=1..19); # _Peter Luschny_, Jun 11 2016

%t Rest[CoefficientList[Series[Log[(1 + 1/Sqrt[1 - 4 x])/2]/Sqrt[1 - 4 x], {x, 0, 20}], x] Range[0, 20]!] (* _Vaclav Kotesovec_, Apr 01 2016 *)

%t a[n_] := HarmonicNumber[n] Pochhammer[n, n];

%t Array[a, 20] (* _Jean-François Alcover_, Jun 13 2016 *)

%o (Maxima) a(n):=n!*sum(binomial(2*n,n-k)/k,k,1,n); /* _Vladimir Kruchinin_, Mar 31 2016 */

%o (PARI) a(n) = n!*sum(k=1, n, binomial(2*n,n-k)/k); \\ _Michel Marcus_, Mar 31 2016

%o (PARI) x='x+O('x^44); Vec(serlaplace(log((1 + 1/sqrt(1-4*x))/2)/sqrt(1-4*x))) \\ _Joerg Arndt_, Apr 01 2016

%K easy,nonn

%O 1,2

%A _Leroy Quet_, Dec 26 2000