A093344 a(n) = n! * Sum_{i=1..n} (1/i)*Sum_{j=0..i-1} 1/j!.

0, 1, 4, 17, 84, 485, 3236, 24609, 210572, 2004749, 21033900, 241237001, 3003349124, 40345599957, 581765196884, 8963453118065, 146969877361116, 2555361954692189, 46963373856864092, 909707559383702169, 18524816853636447380, 395634467245613474981

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..400

Formula

E.g.f.: exp(1)*(Ei(1,1-x)-Ei(1,1))/(1-x). - Vladeta Jovovic, May 05 2007

a(n) = Integral_{x=1..oo} exp(1-x)*(x^n*log(x) - n!/x) dx. - Groux Roland, Mar 12 2011

From Vladimir Reshetnikov, Oct 28 2015: (Start)

a(n) = exp(1)*(H(n)*n! + (Ei(-1)-gamma)*n! + hypergeom([n+1,n+1],[n+2,n+2],-1)/(n+1)^2), where H(n)*n! = A000254(n), -Ei(-1) is A099285, gamma is A001620.

Recurrence: a(0) = 0, a(1) = 1, a(2) = 4, a(n) = 2*n*a(n-1) + (2-n^2)*a(n-2) + (n-2)^2*a(n-3).

(End)

a(n) = n!*e*Sum_{k=1..n} Gamma(k,1)/k!. - Robert Israel, Oct 28 2015

Maple

f:= gfun:-rectoproc({a(0) = 0, a(1) = 1, a(2) = 4, a(n) = 2*n*a(n-1) + (2-n^2)*a(n-2) + (n-2)^2*a(n-3)}, a(n), remember):
seq(f(n), n=0..50); # Robert Israel, Oct 28 2015

Mathematica

Round@Table[E n! Sum[Gamma[k, 1]/k!, {k, 1, n}], {n, 0, 20}]
Round@Table[E ((HarmonicNumber[n] + ExpIntegralEi[-1] - EulerGamma) n! + HypergeometricPFQ[{n+1, n+1}, {n+2, n+2}, -1]/(n+1)^2), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 28 2015 *)

Prog

(PARI) a(n) = n!*sum(i=1, n, 1/i*sum(j=0, i-1, 1/j!))

Crossrefs

Cf. A000254, A000774.

Equals for n=>1 the row sums of A165674 and A093905. - Johannes W. Meijer, Oct 16 2009

Sequence in context: A369476 A200716 A093904 * A087316 A104979 A081052

Adjacent sequences: A093341 A093342 A093343 * A093345 A093346 A093347

Keyword

nonn

Author

Ralf Stephan, Apr 26 2004

Status

approved