A318640 - OEIS

%I #12 Nov 09 2018 18:24:58

%S 1,8,64,704,8704,113536,1544192,21671936,311468032,4560963584,

%T 67807363072,1020767092736,15528671576064,238354043994112,

%U 3686842679427072,57411380912848896,899288363016650752,14160044430295826432,224000813601673707520,3558331523719659257856,56738516167544872632320,907803739948246687023104,14569894558284085117059072,234507206354840677339103232

%N G.f. A(x) satisfies: Sum_{n>=0} n * (x + (-1)^n*A(x))^n = 0.

%H Paul D. Hanna, <a href="/A318640/b318640.txt">Table of n, a(n) for n = 1..500</a>

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

%F (1) A(-A(-x)) = x.

%F (2) 0 = Sum_{n>=0} n * (x + (-1)^n*A)^n.

%F (3) 0 = (A-x)*(1 + (A-x)^2)/(1 - (A-x)^2)^2  -  2*(A+x)^2/(1 - (A+x)^2)^2.

%F (4) 0 = x*(1-x)^2*(1+x)^4 - (1-x)^5*(1+x)*A + (2+x+4*x^2-3*x^3)*(1+x)^2*A^2 + (1-x)^3*(1+3*x)*A^3 - (4+5*x+2*x^2-3*x^3)*A^4 + (1-x)*(1-3*x)*A^5 + (2-x)*A^6 - A^7.

%F (5) A(x) = D(D(D(D( D(D(D(D(x)))) )))), the 8th iteration of the g.f. D(x) of A318643.

%F a(n) ~ c * d^n / n^(3/2), where d = 17.1575459832392661371657069324318450352851378685670176577789845392153106... and c = 0.0649898418070562963132195090430418977694503433390371091160871400852... - _Vaclav Kotesovec_, Sep 06 2018

%e G.f.: A(x) = x + 8*x^2 + 64*x^3 + 704*x^4 + 8704*x^5 + 113536*x^6 + 1544192*x^7 + 21671936*x^8 + 311468032*x^9 + 4560963584*x^10 + ...

%e such that

%e 0 = (x - A(x)) + 2*(x + A(x))^2 + 3*(x - A(x))^3 + 4*(x + A(x))^4 + 5*(x - A(x))^5 + 6*(x + A(x))^6 + 7*(x - A(x))^7 + 8*(x + A(x))^8 + 9*(x - A(x))^9 + 10*(x + A(x))^10 + ...

%e RELATED SERIES.

%e (a) If B(B(x)) = A(x) then

%e B(x) = x + 4*x^2 + 16*x^3 + 160*x^4 + 1408*x^5 + 13760*x^6 + 140288*x^7 + 1459200*x^8 + 15595520*x^9 + 168584192*x^10 + 1847791616*x^11 + ... + A318641(n)*x^n + ...

%e where B(-B(-x)) = x.

%e (b) If C(C(C(C(x)))) = A(x), so that C(C(x)) = B(x), then

%e C(x) = x + 2*x^2 + 4*x^3 + 56*x^4 + 304*x^5 + 2944*x^6 + 22592*x^7 + 196864*x^8 + 1700352*x^9 + 14416896*x^10 + 127798272*x^11 + 1141090304*x^12 + ... + A318642(n)*x^n + ...

%e where C(-C(-x)) = x.

%e (c) If D(D(D(D( D(D(D(D(x)))) )))) = A(x), so that D(D(x)) = C(x), then

%e D(x) = x + x^2 + x^3 + 25*x^4 + 73*x^5 + 1025*x^6 + 4913*x^7 + 48985*x^8 + 311305*x^9 + 2393953*x^10 + 17903761*x^11 + 140986201*x^12 + 1096160649*x^13 + ... + A318643(n)*x^n + ...

%e where D(-D(-x)) = x.

%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = polcoeff(sum(m=1, #A, m*(x + (-1)^m*x*Ser(A))^m), #A)); A[n]}

%o for(n=1, 30, print1(a(n), ", "))

%Y Cf. A318641, A318642, A318643.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Aug 31 2018