A245313 - OEIS

%I #11 Jul 20 2014 12:48:12

%S 1,1,2,7,31,176,1158,8919,76751,742597,7865088,91553100,1150905332,

%T 15665172108,227991734414,3554320236911,58795765799791,

%U 1033303679424539,19151079894682674,374662948814998855,7691131223011551255,165785969673935575904,3734170668741419488552

%N G.f. satisfies: A(x) = 1 + x*A(x) + x^2*A(x)^2 + 3*x^3*A(x)*A'(x) + x^4*A(x)*A''(x).

%C Do the following limits exist? If so, what are the respective values?

%C (1) limit a(n)*sqrt(n+1)/(n+1)! ? (Value is near 0.718490 at n=400.)

%C (2) limit A245313(n)/A245312(n) ? (Value is near 2.721747 at n=400.)

%C Limit a(n)*sqrt(n+1)/(n+1)! = 0.7189460513696389360211370..., limit A245313(n)/A245312(n) = e. - _Vaclav Kotesovec_, Jul 20 2014

%H Paul D. Hanna, <a href="/A245313/b245313.txt">Table of n, a(n) for n = 0..400</a>

%F G.f. A(x) satisfies:

%F (1) A(x) = 1/(1-x - x^2*Dx^2(A(x))) where Dx(F(x)) = d/dx x*F(x).

%F (2) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A245312.

%F (3) A(x) = (1/x)*Series_Reversion(x/G(x)) where G(x) is the g.f. of A245312.

%F a(n) = [x^n] G(x)^(n+1)/(n+1) for n>=0 where G(x) is the g.f. of A245312.

%F a(n) = [x^(n+2)] G(x)^(n+1)/(n+1)^3 for n>=0 where G(x) is the g.f. of A245312.

%e G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 31*x^4 + 176*x^5 + 1158*x^6 +...

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

%e Define Dx(A(x)) = d/dx x*A(x) = A(x) + x*A'(x):

%e Dx(A(x)) = 1 + 2*x + 6*x^2 + 28*x^3 + 155*x^4 + 1056*x^5 + 8106*x^6 +...

%e so that Dx^2(A(x)) = d/dx x*(d/dx x*A(x)) = A(x) + 3*x*A'(x) + x^2*A''(x):

%e Dx^2(A(x)) = 1 + 4*x + 18*x^2 + 112*x^3 + 775*x^4 + 6336*x^5 +...

%e then A(x) = 1/(1-x - x^2*Dx^2(A(x))):

%e 1/A(x) = 1 - x - x^2 - 4*x^3 - 18*x^4 - 112*x^5 - 775*x^6 - 6336*x^7 -...

%e RELATED SERIES.

%e The g.f. of A245312 begins:

%e G(x) = 1 + x + x^2 + 3*x^3 + 10*x^4 + 60*x^5 + 360*x^6 + 2940*x^7 +...

%e where A(x) = G(x*A(x)) where G(x) = A(x/G(x)) and

%e a(n) = [x^n] G(x)^(n+1)/(n+1) = [x^(n+2)] G(x)^(n+1)/(n+1)^3 for n>=0.

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*A+x^2*A^2+3*x^3*A*A'+x^4*A*A''+x*O(x^n));polcoeff(A,n)}

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

%o (PARI) /* From A(x) = G(x*A(x)) where G(x) is the g.f. of A245312: */

%o {a(n)=local(A=[1,1]);for(i=1,n,A=concat(A,0);m=#A-2;A[#A]=-Vec(Ser(A)^m*(1-m^2*x^2))[#A]/m);

%o polcoeff(1/x*serreverse(x/Ser(A)),n)}

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

%Y Cf. A245312, A245311.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 19 2014