A193161 E.g.f. A(x) satisfies: A(x/(1-x))/(1-x) = d/dx x*A(x).

1, 1, 3, 17, 152, 1944, 33404, 738212, 20316288, 679237248, 27050017152, 1262790237312, 68193683598336, 4212508572109824, 294822473048043264, 23184842446161993984, 2033884583922970558464, 197767395237549512097792, 21194678534706844531458048

Offset: 0

Author

Paul D. Hanna, Jul 16 2011

Keywords

• nonn

Comments

In Cellarosi and Sinai (2011) on page 257, m_k denotes a(k)/k!. - Michael Somos, Dec 28 2012

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 152*x^4/4! + 1944*x^5/5! + ...
Related expansions:
A(x/(1-x))/(1-x) = 1 + 2*x + 9*x^2/2! + 68*x^3/3! + 760*x^4/4! + ...
A(x) + x*A'(x) = 1 + 2*x + 9*x^2/2! + 68*x^3/3! + 760*x^4/4! + ...
Also, a(n) appears in the expansion:
B(x) = 1 + x + 3*x^2/2!^2 + 17*x^3/3!^2 + 152*x^4/4!^2 + 1944*x^5/5!^2 + ...
where
log(B(x)) = x + x^2/(2*2!) + x^3/(3*3!) + x^4/(4*4!) + x^5/(5*5!) + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..293
• F. Cellarosi and Ya. G. Sinai, The Möbius function and statistical mechanics, Bull. Math. Sci., 2011.

Crossrefs

Cf. A023998, A193160, A209884.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-i)*binomial(n-1, i-1)/i, i=1..n))
    end:
a:= n-> b(n)*n!:
seq(a(n), n=0..25);  # Alois P. Heinz, May 11 2016

Mathematica

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 *)
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 *)
Table[Sum[BellY[n, k, 1/Range[n]], {k, 0, n}] n!, {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

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)}

PARI

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

PARI

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

Formula

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

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

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

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