A068102 a(n) = n! * 2^n * Sum_{i=1..n} 1/(i*2^i).

1, 5, 32, 262, 2644, 31848, 446592, 7150512, 128749536, 2575353600, 56661408000, 1359913708800, 35358235430400, 990036819072000, 29701191750451200, 950439443688806400, 32314962008209305600, 1163338987982963097600, 44206887945726303436800, 1768275639474152546304000

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Formula

E.g.f.: -log(1-x)/(1-2*x). - Vladeta Jovovic, Feb 07 2003

a(n+1) = 2*(n+1)*a(n) + n!, a(0)=0. - Jaume Oliver Lafont, Sep 15 2009

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

a(n) = (3*n-1)*a(n-1) - 2*(n-1)^2*a(n-2). - Vaclav Kotesovec, Aug 13 2013

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

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

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

Maple

seq(add(n!/i*2^(n-i), i=1..n), n=1..100); # Robert Israel, Aug 14 2014

Mathematica

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

Prog

(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
(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
(Maxima)
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 */

Crossrefs

Cf. A000254, A069015, A087547, A000165, A001008

Sequence in context: A198598 A369785 A215916 * A166993 A328055 A265130

Adjacent sequences: A068099 A068100 A068101 * A068103 A068104 A068105

Keyword

nonn,easy

Author

Benoit Cloitre, Apr 14 2002

Status

approved