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a(n) = n! * 2^n * Sum_{i=1..n} 1/(i*2^i).
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%I #62 Feb 04 2023 14:13:16

%S 1,5,32,262,2644,31848,446592,7150512,128749536,2575353600,

%T 56661408000,1359913708800,35358235430400,990036819072000,

%U 29701191750451200,950439443688806400,32314962008209305600,1163338987982963097600,44206887945726303436800,1768275639474152546304000

%N a(n) = n! * 2^n * Sum_{i=1..n} 1/(i*2^i).

%H Vincenzo Librandi, <a href="/A068102/b068102.txt">Table of n, a(n) for n = 1..200</a>

%F E.g.f.: -log(1-x)/(1-2*x). - _Vladeta Jovovic_, Feb 07 2003

%F a(n+1) = 2*(n+1)*a(n) + n!, a(0)=0. - _Jaume Oliver Lafont_, Sep 15 2009

%F a(n) = 2^n*n!*(log(2) - 2*Integral_{x=0..1} x^(2*n+1)/(1+x^2)^(n+1) dx). Thus a(n)/(2^n*n!) -> log(2) as n -> inf. Cf. A087547. - _Peter Bala_, Jun 21 2013

%F a(n) = (3*n-1)*a(n-1) - 2*(n-1)^2*a(n-2). - _Vaclav Kotesovec_, Aug 13 2013

%F The sequence b(n) = 2^n*n! = A000165(n) also satisfies the above second-order recurrence of Kotesovec. This leads to the generalized continued fraction expansion lim_{n->oo} a(n)/b(n) = log(2) = 1/(2 - 2/(5 - 8/(8 - 18/(11 - ... - 2*(n - 1)^2/((3*n - 1) - ... ))))). - _Peter Bala_, Feb 18 2015

%F a(n)/n! is the linear term of the sum of the n-th row of a Pascal-like triangle T in which T(n,k) = binomial(x+n, k). - _Greg Dresden_ and _Ivan Kuznetsov_, Aug 22 2022

%F a(n) = n!*Sum_{k=floor(n/2)..n} (-1)^(n-k)*C(k,n-k)*C(2*k,k)*(H(2*k)-H(k)), where H(n) are the harmonic numbers. - _Vladimir Kruchinin_, Feb 04 2023

%p seq(add(n!/i*2^(n-i), i=1..n), n=1..100); # _Robert Israel_, Aug 14 2014

%t a[n_] := FullSimplify[n! (2^n Log[2] - LerchPhi[1/2, 1, 1 + n]/2)]; Array[a,10] (* _Vladimir Reshetnikov_, Jan 21 2011 *)

%o (Magma) I:=[1,5]; [n le 2 select I[n] else (3*n-1)*Self(n-1)-2*(n-1)^2*Self(n-2): n in [1..25] ]; // _Vincenzo Librandi_, Feb 19 2015

%o (PARI) first(n)=my(v=vector(n),t=1); v[1]=1; for(k=2,n, v[k]=2*k*v[k-1] + t; t*=k); v \\ _Charles R Greathouse IV_, Aug 22 2022

%o (Maxima)

%o a(b):=n!*sum(binomial(k,n-k)*(-1)^(n-k)*binomial(2*k,k)*(H(2*k)-H(k)),k,floor(n/2),n); /* _Vladimir Kruchinin_, Feb 04 2023 */

%Y Cf. A000254, A069015, A087547, A000165, A001008

%K nonn,easy

%O 1,2

%A _Benoit Cloitre_, Apr 14 2002