A245308 - OEIS

%I #14 Jul 24 2014 15:24:28

%S 1,1,3,19,205,3501,90271,3357103,171841209,11598601465,996140770651,

%T 105829573610091,13602095395648453,2077762791361106149,

%U 371766799417828843575,76978381709312988826951,18256702588619162109630961,4915636696257611754342845553,1491009565882345791444427756339

%N E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * ( d/dx x*A(x)^n ) / A(x)^n.

%H Vaclav Kotesovec, <a href="/A245308/b245308.txt">Table of n, a(n) for n = 0..150</a>

%F E.g.f. satisfies:

%F (1) A(x) = exp(x) * (1 + x^2*A'(x)/A(x)).

%F (2) A(x) = exp( Sum_{n>=1} A245119(n+1)*x^n/n ).

%F a(n) ~ c * (n!)^2 / n, where c = BesselJ(1,2) = 0.5767248077568733872... . - _Vaclav Kotesovec_, Jul 22 2014

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 205*x^4/4! + 3501*x^5/5! +...

%e such that A(x) = exp(x) * (1 + x^2*A'(x)/A(x)) where

%e 1 + x^2*A'(x)/A(x) = 1 + 2*x^2/2! + 12*x^3/3! + 144*x^4/4! + 2640*x^5/5! + 72000*x^6/6! + 2792160*x^7/7! + 147329280*x^8/8! +...

%e RELATED SERIES.

%e The e.g.f. equals the product of exp(x) and an integer series (A245119):

%e exp(-x)*A(x) = 1 + x^2 + 2*x^3 + 6*x^4 + 22*x^5 + 100*x^6 + 554*x^7 + 3654*x^8 + 28014*x^9 + 244572*x^10 + 2392042*x^11 + 25877610*x^12 +...+ A245119(n)*x^n +...

%e The logarithmic derivative of the e.g.f. is an integer series:

%e A'(x)/A(x) = 1 + 2*x + 6*x^2 + 22*x^3 + 100*x^4 + 554*x^5 + 3654*x^6 + 28014*x^7 + 244572*x^8 + 2392042*x^9 + 25877610*x^10 +...+ A245119(n+2)*x^n +...

%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=sum(m=0,n,x^m*deriv(x*A^m)/A^m/m!+x*O(x^n)));n!*polcoeff(A,n)}

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

%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=sum(m=0,n,x^m*(1+m*x*A'/A)/m!+x*O(x^n)));n!*polcoeff(A,n)}

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

%Y Cf. A245119.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 21 2014