A193332 E.g.f. satisfies: A(x) = x*exp( A(x)/A'(x) ).

1, 2, -3, 52, -1315, 50286, -2655863, 183322952, -15928677063, 1695597280570, -216636191518219, 32688113040335292, -5749136647259226923, 1165789270581830003942, -270019628802455686919295, 70862777375461690495134736, -20921819854506620454336189583

Offset: 1

Author

Paul D. Hanna, Jul 23 2011

Keywords

• sign

Examples

			E.g.f.: A(x) = x + 2*x^2/2! - 3*x^3/3! + 52*x^4/4! - 1315*x^5/5! + 50286*x^6/6! - 2655863*x^7/7! + 183322952*x^8/8! +...
where A(x)/A'(x) = log(A(x)/x) equals the integer series:
(1) A(x)/A'(x) = x - x^2 + 3*x^3 - 14*x^4 + 85*x^5 - 621*x^6 + 5236*x^7 - 49680*x^8 + 521721*x^9 - 5994155*x^10 +...
which equals -G(-x) where G(x) is the g.f. of A088716.
The series reversion, -L(-x), begins:
(2) -L(-x) = x - 2*x^2/2! + 15*x^3/3! - 232*x^4/4! + 5905*x^5/5! - 220176*x^6/6! + 11210479*x^7/7! - 743759360*x^8/8! +...
where L(x) is the e.g.f. A161968.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..250

Crossrefs

Cf. A088716, A161968, A161967.

Programs

PARI

{a(n)=local(A=x+x^2);for(i=1,n,A=x*exp(A/(A'+x*O(x^n))));n!*polcoeff(A,n)}

Formula

E.g.f. A(x) satisfies:

(1) A(x)/A'(x) = -G(-x) where G(x) = x + x*G(x)*G'(x) is a g.f. of A088716; thus, log(A(x)/x) is an integer series.

(2) A(-L(-x)) = x where L(x) = x*exp(x*L'(x)) is the e.g.f. of A161968.

a(n) ~ c * (-1)^n * (n!)^2, where c = 0.217950789447151065... (see A238223). - Vaclav Kotesovec, Feb 26 2014