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A058806 a(n) = n! * H_n(n) where H_0(n) = 1/n, H_m(n) = Sum_{k=1..n} H_{m-1}(k).

%I A058806 #37 Mar 12 2025 04:45:57
%S A058806 1,5,47,638,11274,245004,6314664,188204400,6366517200,240947474400,
%T A058806 10086271796160,462688566802560,23080457713017600,1243853764482470400,
%U A058806 72018614888670643200,4458392682933188966400,293860908364035250022400,20545850809171272549888000,1518779004111434057997312000
%N A058806 a(n) = n! * H_n(n) where H_0(n) = 1/n, H_m(n) = Sum_{k=1..n} H_{m-1}(k).
%H A058806 John Tyler Rascoe, <a href="/A058806/b058806.txt">Table of n, a(n) for n = 1..100</a>
%F A058806 a(1) = 1; a(n) = 2*(2*n-1)*a(n-1) - (2*n-3)!/(n-1)!.
%F A058806 a(n) = (2*n)!/(4*n!)*(Psi(n+1/2) - Psi(n) + 2*log(2)). - _Vladeta Jovovic_, Jan 22 2005
%F A058806 E.g.f.: log((sqrt(1-4*x)+1)/2)*(2*x-sqrt(1-4*x)-1)/(-4*x+sqrt(1-4*x)+1). - _Vladimir Kruchinin_, Mar 17 2016
%F A058806 a(n) = n!*Sum_{k=1..n} binomial(2*n-k-1, n-k)/k. - _Vladimir Kruchinin_, Mar 17 2016
%F A058806 a(n) ~ log(2) * 2^(2*n - 1/2) * n^n / exp(n). - _Vaclav Kotesovec_, Mar 17 2016
%F A058806 a(n) = n! * [x^n] -log(1 - x)/(1 - x)^n. - _Ilya Gutkovskiy_, Sep 21 2017
%F A058806 From _Peter Bala_, Mar 07 2025: (Start)
%F A058806 a(n+1) = - (2*n+1)!^2/n!^3 * Integral_{x = 0..1} log(x)*x^n*(1 - x)^n = (2*n+1)!^2/n!^3 * Sum_{k = 0..n} (-1)^k * binomial(n, k)/(n+k+1)^2.
%F A058806 (n-1)*a(n) = 2*(4*n^2-10*n+7)*a(n-1) - 4*(n-2)*(2*n-3)^2*a(n-2) with a(1) = 1, a(2) = 5. (End)
%e A058806 a(3) = 3!*(1 + (1 + (1 + 1/2)) + (1 + (1 + 1/2) + (1 + 1/2 + 1/3))) = 47.
%p A058806 a := proc(n) option remember; if n = 1 then 1 else 2*(2*n-1)*a(n-1) - (2*n-3)!/(n-1)! fi; end:
%p A058806 seq(a(n), n = 1..20); # _Peter Bala_, Mar 07 2025
%t A058806 Table[n! Sum[Binomial[2 n - k - 1, n - k]/k, {k, n}], {n, 19}] (* _Michael De Vlieger_, Mar 17 2016 *)
%o A058806 (Maxima)
%o A058806 a(n):=n!*sum(binomial(2*n-k-1,n-k)/k,k,1,n);
%o A058806 /* _Vladimir Kruchinin_, Mar 17 2016 */
%o A058806 (PARI) lista(nn) = {print1(a=1, ", "); for (n=2, nn, a = a*2*(2*n-1) - (2*n-3)!/(n-1)!; print1(a, ", "););} \\ _Michel Marcus_, Mar 17 2016
%Y A058806 Cf. A000108.
%K A058806 easy,nonn
%O A058806 1,2
%A A058806 _Leroy Quet_, Jan 02 2001
%E A058806 More terms from _Michael De Vlieger_, Mar 17 2016