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 A245309 E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * (d/dx x*A(x)^n)^n / A(x)^(n^2). 1

%I

%S 1,1,3,25,397,10101,372991,18744853,1227094905,101320257097,

%T 10294575759451,1262050509059121,183700770307306693,

%U 31322680620408105085,6184922808789945458967,1400325997347499801032301,360395936189117983848624241,104632853179210298481432557073

%N E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * (d/dx x*A(x)^n)^n / A(x)^(n^2).

%H Vaclav Kotesovec, <a href="/A245309/b245309.txt">Table of n, a(n) for n = 0..130</a>

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

%F (1) A(x) = Sum_{n>=0} x^n * (1 + n*x*A'(x)/A(x))^n / n!.

%F (2) A(x) = Sum_{n>=0} x^n * Sum_{k=0..[n/2]} C(n-k,k) * (n-k)^k * A'(x)^k/A(x)^k / (n-k)!.

%F a(n) ~ c * (n!)^2, where c = 0.881770167... . - _Vaclav Kotesovec_, Jul 25 2014

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 25*x^3/3! + 397*x^4/4! + 10101*x^5/5! +...

%e where

%e A(x) = 1 + x*(d/dx x*A(x))/A(x) + x^2/2!*(d/dx x*A(x)^2)^2/A(x)^4 + x^3/3!*(d/dx x*A(x)^3)^3/A(x)^9 + x^4/4!*(d/dx x*A(x)^4)^4/A(x)^16 +...

%e or, equivalently,

%e A(x) = 1 + x*(1 + x*A'(x)/A(x)) + x^2*(1 + 2*x*A'(x)/A(x))^2/2! + x^3*(1 + 3*x*A'(x)/A(x))^3/3! + x^4*(1 + 4*x*A'(x)/A(x))^4/4! +...

%e Related series:

%e A'(x)/A(x) = 1 + 2*x + 18*x^2/2! + 300*x^3/3! + 7980*x^4/4! + 305520*x^5/5! + 15801240*x^6/6! + 1058302560*x^7/7! + 88992343440*x^8/8! +...

%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)^m/A^(m^2)/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/m!+x*O(x^n)));n!*polcoeff(A,n)}

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

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 23 2014

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Last modified May 17 19:51 EDT 2022. Contains 353778 sequences. (Running on oeis4.)