A073478 Expansion of (1+x)^(1/(1-x)).

1, 1, 2, 9, 44, 290, 2154, 19026, 186752, 2070792, 25119720, 334960560, 4824346152, 75100568088, 1250180063664, 22235660291880, 419595248663040, 8388866239417920, 176823515257447104, 3923498370610292544

Offset: 0

Author

Vladeta Jovovic, Aug 26 2002

Keywords

• nonn

Examples

			E.g.f.: (1+x)^(1/(1-x)) = 1 + x + 2*x^2/2! + 9*x^3/3! + 44*x^4/4! + 290*x^5/5! + 2154*x^6/6! + 19026*x^7/7! + 186752*x^8/8! + 2070792*x^9/9! + ...
which may be written as
(1+x)^(1/(1-x)) = exp(x + x^2*(1+x)/2 + x^3*(1+x+x^2)/3 + x^4*(1+x+x^2+x^3)/4 + x^5*(1+x+x^2+x^3+x^4)/5 + ... + x^n*((1-x^n)/(1-x))/n + ...).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A007113, A007120, A087761.

Programs

Mathematica

CoefficientList[Series[(1+x)^(1/(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Apr 21 2014 *)

PARI

{a(n)=n!*polcoeff((1+x +x*O(x^n))^(1/(1-x)),n)} \\ Paul D. Hanna, Jan 08 2014

PARI

{a(n)=local(A);A=exp(sum(m=1,n,sum(k=1,m,-(-1)^k/k)*x^m)+x*O(x^n)); n!*polcoeff(A,n)} \\ Paul D. Hanna, Jan 08 2014

Formula

E.g.f.: exp( Sum_{n>=1} x^n * Sum_{k=1..n} -(-1)^k/k ). - Paul D. Hanna, Jan 08 2014

E.g.f.: exp( Sum_{n>=1} x^n * ((1-x^n)/(1-x)) / n ). - Paul D. Hanna, Nov 24 2024

a(n) ~ (log(2))^(1/4) * exp(2*sqrt(n*log(2)) - n - 1/2) * n^(n-1/4). - Vaclav Kotesovec, Apr 21 2014

Extensions

More terms from Robert G. Wilson v, Aug 28 2002