A193425 Expansion of e.g.f.: (1 - 2*x)^(-1/(1-x)).

1, 2, 12, 96, 976, 12000, 172608, 2838528, 52474112, 1076451840, 24254069760, 595235266560, 15801350443008, 451082627014656, 13778232107286528, 448348123661598720, 15483358506138009600, 565560454279135887360

Offset: 0

Author

Paul D. Hanna, Jul 27 2011

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 96*x^3/3! + 976*x^4/4! + 12000*x^5/5! +...
where the logarithm involves sums of reciprocal binomial coefficients:
log(A(x)) = 2*x*(1) + (2*x)^2/2*(1 + 1) + (2*x)^3/3*(1 + 1/2 + 1) + (2*x)^4/4*(1 + 1/3 + 1/3 + 1) + (2*x)^5/5*(1 + 1/4 + 1/6 + 1/4 + 1) + (2*x)^6/6*(1 + 1/5 + 1/10 + 1/10 + 1/5 + 1) +...
Explicitly, the logarithm begins:
log(A(x)) = 2*x + 8*x^2/2! + 40*x^3/3! + 256*x^4/4! + 2048*x^5/5! + 19968*x^6/6! +...
in which the coefficients equal 2*A126674(n).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A126674.

Programs

Magma

m:=50;
f:= func< x | Exp((&+[(&+[ 1/Binomial(n-1,k): k in [0..n-1]])*(2*x)^n/n: n in [1..m+2]])) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Feb 02 2023

Mathematica

CoefficientList[Series[(1-2*x)^(-1/(1-x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *)

PARI

{a(n)=n!*polcoeff(exp(sum(m=1,n,2^m*x^m/m*sum(k=0,m-1,1/binomial(m-1,k)))+x*O(x^n)),n)}

PARI

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

SageMath

m=50
def f(x): return exp(sum(sum( 1/binomial(n-1,k) for k in range(n))*(2*x)^n/n for n in range(1,m+2)))
def A193425_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).egf_to_ogf().list()
A193425_list(m) # G. C. Greubel, Feb 02 2023

Formula

E.g.f.: exp( Sum_{n>=1} (2*x)^n/n * Sum_{k=0..n-1} 1/C(n-1,k) ).

E.g.f.: exp( Sum_{n>=1} 2*A126674(n)*x^n/n ), where A126674(n) = n!*Sum_{j=0..n-1} 2^j/(j+1).

a(n) ~ n!*n*2^n * (1 - 2*log(n)/n). - Vaclav Kotesovec, Jun 27 2013