A103213 a(n) = n! * Sum_{k=1..n} binomial(n,k)/k.

1, 5, 29, 206, 1774, 18204, 218868, 3036144, 47928816, 850514400, 16783812000, 364865040000, 8666747625600, 223351748524800, 6206847295622400, 185007996436838400, 5887506932836300800, 199216094254423142400

Offset: 1

Author

Ralf Stephan, Jan 28 2005

Keywords

• nonn

Comments

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

Links

• Robert Israel, Table of n, a(n) for n = 1..369

Crossrefs

Cf. A021009.

Programs

Maple

S:= series(log((1-2*x)/(1-x))/(x-1), x, 41):
seq(coeff(S,x,j)*j!,j=1..40); # Robert Israel, Jun 19 2015

Mathematica

a[n_] := n*n!*HypergeometricPFQ[{1, 1, 1-n}, {2, 2}, -1]; Table[a[n], {n, 1, 18}] (* Jean-François Alcover, Nov 28 2013 *)
Table[n! (-HarmonicNumber[n] - 2^(n+1) Re[LerchPhi[2, 1, n+1]]), {n, 1, 20}] (* Vladimir Reshetnikov, Apr 24 2016 *)

Formula

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

a(n) ~ 2^(n+1)*(n-1)!. - Jean-François Alcover, Nov 28 2013

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

a(n) = -n!*log(2) + Sum_{k>=1} (k+n)!/(2^k*k*k!). - Groux Roland, Dec 18 2010

From Vladimir Reshetnikov, Apr 24 2016: (Start)

a(n) = n!*((n+1)*hypergeom([1, 1, n+2], [2, 2], 1/2)/2 - log(2)).

a(n) = n!*(-H(n) - Re(Beta(2; n+1, 0))).

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.

(End)

a(n) = -Sum_{k=0..n} (-1)^k*k!*A021009(n, k+1). - Mélika Tebni, Jun 20 2022