A209884 -id:A209884 - cp's OEIS frontend

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

Paul D. Hanna, Jul 16 2011

nonn

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

			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!) + ...

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.

Cf. A023998, A193160, A209884.

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

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 *)

{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)}

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

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

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).

A209917 E.g.f. A(x) satisfies: A(x/(1-x))/(1-x) = (1/x^2) * d/dx x^3*A(x)/3.

Original entry on oeis.org

1, 3, 21, 249, 4356, 103932, 3213216, 124146432, 5834291328, 326570493312, 21408981213888, 1621281984076224, 140205279698051328, 13711076231477352192, 1503606581609959001088, 183562416179374733411328, 24787906630769478567297024

Offset: 0

Paul D. Hanna, Mar 15 2012

nonn

			E.g.f.: A(x) = 1 + 3*x + 21*x^2/2! + 249*x^3/3! + 4356*x^4/4! + 103932*x^5/5! + ...
Related expansions:
A(x/(1-x))/(1-x) = 1 + 4*x + 35*x^2/2! + 498*x^3/3! + 10164*x^4/4! + ...
A(x) + x*A'(x)/3 = 1 + 4*x + 35*x^2/2! + 498*x^3/3! + 10164*x^4/4! + ...
Also, a(n) appears in the expansion:
B(x) = 1 + 3*x + 21*x^2/2!^2 + 249*x^3/3!^2 + 4356*x^4/4!^2 + 103932*x^5/5!^2 + ...
such that
log(B(x)) = 3*x + 3*x^2/(2*2!) + 3*x^3/(3*3!) + 3*x^4/(4*4!) + 3*x^5/(5*5!) + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A193161, A209884.

a = ConstantArray[0,21]; a[[1]]=1; a[[2]]=3; Do[a[[n+1]] = (n-1)!*Sum[3*Binomial[n, k]*a[[k+1]]/k!,{k,0,n-1}],{n,2,20}]; a  (* Vaclav Kotesovec, Feb 23 2014 *)
Table[Sum[BellY[n, k, 3/Range[n]], {k, 0, n}] n!, {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

{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+3*intformal((B-A)/x)); n!*polcoeff(A, n)}

{a(n)=if(n<0, 0, if(n==0, 1, (n-1)!*sum(k=0, n-1, 3*binomial(n, k)*a(k)/k!)))}

{a(n)=n!^2*polcoeff(exp(sum(m=1, n, 3*x^m/(m*m!))+x*O(x^n)), n)}
for(n=0,30,print1(a(n),", "))

@CachedFunction
def a(n): return 1 if (n==0) else factorial(n-1)*sum( 3*binomial(n, j)*a(j)/factorial(j) for j in (0..n-1) )
[a(n) for n in (0..20)] # G. C. Greubel, Jun 23 2021

E.g.f.: exp( Sum_{n>=1} 3*x^n/(n*n!) ) = Sum_{n>=0} a(n)*x^n/n!^2.

a(n) = (n-1)!* Sum_{k=0..n-1} 3*binomial(n,k)*a(k)/k! for n>0 with a(0)=1.

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

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A209917 E.g.f. A(x) satisfies: A(x/(1-x))/(1-x) = (1/x^2) * d/dx x^3*A(x)/3.

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