A203852 Expansion of e.g.f. exp( Integral -log(1-x) dx ).

1, 0, 1, 1, 5, 16, 79, 421, 2673, 19216, 156021, 1411873, 14117773, 154730720, 1845959179, 23826445501, 330951133537, 4923574598112, 78123812086441, 1317174439409409, 23517962293307701, 443340968936640496, 8799729204814165223, 183448995762912568885

Offset

0,5

Comments

Row sums of A185164. - Peter Bala, Mar 14 2012

Links

G. C. Greubel, Table of n, a(n) for n = 0..450

Formula

E.g.f.: exp( Sum_{n>=2} x^n/(n*(n-1)) ).

E.g.f.: exp(x)*(1-x)^(1-x). - Vaclav Kotesovec, Dec 27 2013

a(n) ~ (n-1)! * (exp(1)/n + (2*log(n)+2*gamma)/n^2), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Dec 27 2013

a(n) = sum(i=1..n-1, binomial(n-1,i)*(i-1)!*a(n-i-1),i,1,n-1), a(0)=1. - Vladimir Kruchinin, Feb 23 2015

Example

E.g.f.: A(x) = 1 + x^2/2! + x^3/3! + 5*x^4/4! + 16*x^5/5! + 79*x^6/6! +...
where: log(A(x)) = x^2/2 + x^3/6 + x^4/12 + x^5/20 + x^6/30 + x^7/42 +...

Mathematica

CoefficientList[Series[Exp[x]*(1-x)^(1-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 27 2013 *)

Prog

(PARI) {a(n)=n!*polcoeff(exp(-intformal(log(1-x +x*O(x^n)))), n)}
(PARI) x='x+O('x^30); Vec(serlaplace(exp(x)*(1-x)^(1-x))) \\ G. C. Greubel, Jul 17 2018
(Maxima)
a(n):=if n=0 then 1 else sum(binomial(n-1, i)*(i-1)!*a(n-i-1), i, 1, n-1); /* Vladimir Kruchinin, Feb 23 2015 */
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x)*(1-x)^(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 17 2018

Crossrefs

Cf. A185164.

Sequence in context: A363558 A197052 A334647 * A179663 A372174 A227292

Adjacent sequences: A203849 A203850 A203851 * A203853 A203854 A203855

Keyword

nonn

Author

Paul D. Hanna, Jan 29 2012

Status

approved