%I #36 Nov 10 2016 03:50:01

%S 1,1,3,17,152,1944,33404,738212,20316288,679237248,27050017152,

%T 1262790237312,68193683598336,4212508572109824,294822473048043264,

%U 23184842446161993984,2033884583922970558464,197767395237549512097792,21194678534706844531458048

%N E.g.f. A(x) satisfies: A(x/(1-x))/(1-x) = d/dx x*A(x).

%C In Cellarosi and Sinai (2011) on page 257, m_k denotes a(k)/k!. - _Michael Somos_, Dec 28 2012

%H Alois P. Heinz, <a href="/A193161/b193161.txt">Table of n, a(n) for n = 0..293</a>

%H F. Cellarosi and Ya. G. Sinai, <a href="http://dx.doi.org/10.1007/s13373-011-0011-6">The Möbius function and statistical mechanics</a>, Bull. Math. Sci., 2011.

%F a(n) = (n-1)!* Sum_{k=0..n-1} binomial(n,k)*a(k)/k! for n>0 with a(0)=1.

%F a(n) = A193160(n+1)/(n+1).

%F E.g.f.: exp( Sum_{n>=1} x^n/(n*n!) ) = Sum_{n>=0} a(n)*x^n/n!^2.

%F a(n) = n! * A177208(n) / A177209(n) for n>=1 (see comment from Michael Somos).

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 152*x^4/4! + 1944*x^5/5! + ...

%e Related expansions:

%e A(x/(1-x))/(1-x) = 1 + 2*x + 9*x^2/2! + 68*x^3/3! + 760*x^4/4! + ...

%e A(x) + x*A'(x) = 1 + 2*x + 9*x^2/2! + 68*x^3/3! + 760*x^4/4! + ...

%e Also, a(n) appears in the expansion:

%e B(x) = 1 + x + 3*x^2/2!^2 + 17*x^3/3!^2 + 152*x^4/4!^2 + 1944*x^5/5!^2 + ...

%e where

%e log(B(x)) = x + x^2/(2*2!) + x^3/(3*3!) + x^4/(4*4!) + x^5/(5*5!) + ...

%p b:= proc(n) option remember; `if`(n=0, 1,

%p add(b(n-i)*binomial(n-1, i-1)/i, i=1..n))

%p end:

%p a:= n-> b(n)*n!:

%p seq(a(n), n=0..25); # _Alois P. Heinz_, May 11 2016

%t a[ n_] := If[ n<0, 0, n!^2 Assuming[ x>0, SeriesCoefficient[ Exp[ Integrate[ (Exp[t] - 1)/t, {t, 0, x}]], {x, 0, n}]]]; (* _Michael Somos_, Dec 28 2012 *)

%t a[ n_] := If[ n<0, 0, n!^2 Assuming[ x>0, SeriesCoefficient[ Exp[ ExpIntegralEi[x] - Log[x] - EulerGamma], {x, 0, n}]]]; (* _Michael Somos_, Dec 28 2012 *)

%t Table[Sum[BellY[n, k, 1/Range[n]], {k, 0, n}] n!, {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 09 2016 *)

%o (PARI) {a(n)=local(A=1+x,B);for(i=1,n,B=subst(A,x,x/(1-x+x*O(x^n)))/(1-x);A=1+intformal((B-A)/x));n!*polcoeff(A,n)}

%o (PARI) {a(n)=if(n<0,0,if(n==0,1,(n-1)!*sum(k=0,n-1,binomial(n,k)*a(k)/k!)))}

%o (PARI) {a(n)=n!^2*polcoeff(exp(sum(m=1,n,x^m/(m*m!))+x*O(x^n)),n)}

%Y Cf. A023998, A193160, A209884.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 16 2011