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%I #38 Jun 20 2022 14:26:59
%S 1,5,29,206,1774,18204,218868,3036144,47928816,850514400,16783812000,
%T 364865040000,8666747625600,223351748524800,6206847295622400,
%U 185007996436838400,5887506932836300800,199216094254423142400
%N a(n) = n! * Sum_{k=1..n} binomial(n,k)/k.
%C a(n) is the sum of all terms in the rows of permutations of the powers of 2. For k=1..n, term(k) can be any power of 2 from 0 to k-1; thus for term(3) it may be 1 or 2 or 4. Find all n! rows of permutations and the sum of the terms in all these rows. This sum will be a(n). - _J. M. Bergot_, Jun 18 2015
%H Robert Israel, <a href="/A103213/b103213.txt">Table of n, a(n) for n = 1..369</a>
%F E.g.f.: log((1-2*x)/(1-x))/(x-1). a(n) = n! * Sum_{k=1..n} (2^k-1)/k. - _Vladeta Jovovic_, Jan 29 2005
%F a(n) ~ 2^(n+1)*(n-1)!. - _Jean-François Alcover_, Nov 28 2013
%F a(n+4) = 2*(n+3)*(n+2)^2*a(n+1)-(n+3)*(13+5*n)*a(n+2)+(4*n+13)*a(n+3). - _Robert Israel_, Jun 19 2015
%F a(n) = -n!*log(2) + Sum_{k>=1} (k+n)!/(2^k*k*k!). - _Groux Roland_, Dec 18 2010
%F From _Vladimir Reshetnikov_, Apr 24 2016: (Start)
%F a(n) = n!*((n+1)*hypergeom([1, 1, n+2], [2, 2], 1/2)/2 - log(2)).
%F a(n) = n!*(-H(n) - Re(Beta(2; n+1, 0))).
%F a(n) = n!*(-H(n) - 2^(n+1)*Re(LerchPhi(2, 1, n+1))), where H(n) is the harmonic number, Beta(z; a, b) is the incomplete Beta function, LerchPhi(z, s, a) is the Lerch transcendent.
%F (End)
%F a(n) = -Sum_{k=0..n} (-1)^k*k!*A021009(n, k+1). - _Mélika Tebni_, Jun 20 2022
%p S:= series(log((1-2*x)/(1-x))/(x-1), x, 41):
%p seq(coeff(S,x,j)*j!,j=1..40); # _Robert Israel_, Jun 19 2015
%t a[n_] := n*n!*HypergeometricPFQ[{1, 1, 1-n}, {2, 2}, -1]; Table[a[n], {n, 1, 18}] (* _Jean-François Alcover_, Nov 28 2013 *)
%t Table[n! (-HarmonicNumber[n] - 2^(n+1) Re[LerchPhi[2, 1, n+1]]), {n, 1, 20}] (* _Vladimir Reshetnikov_, Apr 24 2016 *)
%Y Cf. A021009.
%K nonn
%O 1,2
%A _Ralf Stephan_, Jan 28 2005
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