A268171 E.g.f. A(x) satisfies: A(x) = exp(1+x - exp(x)) * exp( Integral C(x) dx ) such that C(x) = exp( Integral A(x) dx ), where the constant of integration is zero.

1, 1, 1, 2, 9, 46, 245, 1474, 10315, 82174, 726591, 7038632, 74216949, 847103658, 10407684559, 136932309578, 1920656913247, 28609816527534, 451057722743007, 7503877147726572, 131368238149821145, 2414204385183262842, 46469039032487849079, 934915621488296098358, 19624030747998750863203

Offset: 0

Author

Paul D. Hanna, Jan 29 2016

Keywords

• nonn

Comments

Compare to: F(x) = exp( Integral G(x) dx ) such that G(x) = exp(1-exp(x)) * exp( Integral F(x) dx ) holds when F(x) = exp(x).

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 2*x^3/3! + 9*x^4/4! + 46*x^5/5! + 245*x^6/6! + 1474*x^7/7! + 10315*x^8/8! + 82174*x^9/9! + 726591*x^10/10! + 7038632*x^11/11! + 74216949*x^12/12! +...
where
C(x) = 1 + x + 2*x^2/2! + 5*x^3/3! + 16*x^4/4! + 65*x^5/5! + 326*x^6/6! + 1947*x^7/7! + 13410*x^8/8! + 104181*x^9/9! + 900214*x^10/10! + 8566655*x^11/11! +...+ A268170(n)*x^n/n! +...
and A(x) and C(x) satisfy:
(1) A(x) = C'(x)/C(x),
(2) C(x) = A'(x)/A(x) + exp(x) - 1,
(3) log(C(x)) = Integral A(x) dx,
(4) log(A(x)) = Integral C(x) dx + 1+x - exp(x).
		

Programs

PARI

{a(n) = my(A=1+x, C=1+x); for(i=0, n,  A = exp(1+x - exp(x +x*O(x^n))) * exp( intformal( C + x*O(x^n) ) ); C = exp( intformal( A ) );  ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

Logarithmic derivative of A268170.