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

0,4

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).

Links

Table of n, a(n) for n=0..24.

Formula

Logarithmic derivative of A268170.

Example

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).

Prog

(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), ", "))

Crossrefs

Sequence in context: A074607 A251178 A162725 * A168431 A036726 A219197

Adjacent sequences: A268168 A268169 A268170 * A268172 A268173 A268174

Keyword

nonn

Author

Paul D. Hanna, Jan 29 2016

Status

approved