A173895 E.g.f. satisfies: A'(x) = 1/(1 + x*A(x)) with A(0) = 1.

1, 1, -1, 0, 9, -48, 15, 2448, -24927, 23424, 3091311, -47659200, 88056969, 10702667520, -225139993377, 679791291648, 78646340795265, -2128005345251328, 9456106738649631, 1053535684549174272

Offset: 0

Author

Peter Bala, Nov 26 2010

Keywords

• easy
• sign

Comments

Define a polynomial sequence P_n(x) recursively by

... P_0(x) = 1, and for n >= 1

... P_n(x) = (x-1)*P_(n-1)(x-1)-n*P_(n-1)(x+1).

The first few polynomials are

P_1(x) = x-2

P_2(x) = x^2-6*x+5

P_3(x) = x^3-12*x^2+32*x-12.

It appears that a(n+1) = P_n(1) (checked as far as a(19)).

Compare with A144010.

Examples

			E.g.f.: A(x) = 1 + x - x^2/2! + 9*x^4/4! - 48*x^5/5! + 15*x^6/6! + 2448*x^7/7! +...
where
1/(1 + x*A(x)) = 1 - x + 9*x^3/3! - 48*x^4/4! + 15*x^5/5! + 2448*x^6/6! +...
Also, A(G(x)) = 1 + x where
G(x) = x + x^2/2! + 3*x^3/3! + 6*x^4/4! + 18*x^5/5! + 48*x^6/6! + 156*x^7/7! + 492*x^8/8! +...+ A000932(n-1)*x^n/n! +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..190

Crossrefs

Cf. A144010, A000932.

Programs

Mathematica

m = 20; A[_] = 1;
Do[A[x_] = 1 + Integrate[1/(1+x*A[x])+O[x]^m, x]+O[x]^m // Normal, {m}];
CoefficientList[A[x], x] * Range[0, m-1]! (* Jean-François Alcover, Nov 02 2019 *)

PARI

{a(n)=local(A=1+x); for(i=0, n, A=1+intformal(1/(1+x*A+x*O(x^n)) ));n!*polcoeff(A, n)}

Formula

E.g.f. satisfies: A(x) = 1 + Integral 1/(1 + x*A(x)) dx.

E.g.f. satisfies: A(G(x)) = 1 + x where G(x) is the e.g.f. of A000932 (offset 1). [Paul D. Hanna, Aug 23 2011]