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A193161 E.g.f. A(x) satisfies: A(x/(1-x))/(1-x) = d/dx x*A(x). 4
1, 1, 3, 17, 152, 1944, 33404, 738212, 20316288, 679237248, 27050017152, 1262790237312, 68193683598336, 4212508572109824, 294822473048043264, 23184842446161993984, 2033884583922970558464, 197767395237549512097792, 21194678534706844531458048 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

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.

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).

EXAMPLE

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!) + ...

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 *)

PROG

(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)}

CROSSREFS

Cf. A023998, A193160, A209884.

Sequence in context: A075820 A145081 A020562 * A208832 A135751 A168441

Adjacent sequences:  A193158 A193159 A193160 * A193162 A193163 A193164

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 16 2011

STATUS

approved

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Last modified November 21 11:01 EST 2018. Contains 317447 sequences. (Running on oeis4.)