A268171 - OEIS

%I #5 Jan 29 2016 17:56:27

%S 1,1,1,2,9,46,245,1474,10315,82174,726591,7038632,74216949,847103658,

%T 10407684559,136932309578,1920656913247,28609816527534,

%U 451057722743007,7503877147726572,131368238149821145,2414204385183262842,46469039032487849079,934915621488296098358,19624030747998750863203

%N 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.

%C 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).

%F  Logarithmic derivative of A268170.

%e 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! +...

%e where

%e 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! +...

%e and A(x) and C(x) satisfy:

%e (1) A(x) = C'(x)/C(x),

%e (2) C(x) = A'(x)/A(x) + exp(x) - 1,

%e (3) log(C(x)) = Integral A(x) dx,

%e (4) log(A(x)) = Integral C(x) dx + 1+x - exp(x).

%o (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)}

%o for(n=0, 30, print1(a(n), ", "))

%K nonn

%O 0,4

%A _Paul D. Hanna_, Jan 29 2016