A211180 - OEIS

%I #9 May 11 2012 12:37:51

%S 1,1,1,2,3,6,10,21,40,85,170,362,752,1618,3438,7447,16065,35080,76574,

%T 168424,370880,820968,1820598,4052594,9038457,20216002,45301380,

%U 101746560,228918438,516016266,1165005168,2634463663,5965815375,13528669545,30717837778

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

%H Paul D. Hanna, <a href="/A211180/b211180.txt">Table of n, a(n) for n = 1..512</a>

%F Given g.f. A(x), define g(x) = x^2*(1 + A(x)); since A(x) = x + A(g(x))

%F then A(x) equals the sum of all iterations of g(x):

%F A(x) = x + g(x) + g(g(x)) + g(g(g(x))) + g(g(g(g(x)))) +...

%e G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 10*x^7 + 21*x^8 +...

%e Define g(x) = x^2*(1 + A(x)) which satisfies: A(g(x)) = A(x) - x,

%e then the initial iterations of g(x) begin:

%e g(x) = x^2 + x^3 + x^4 + x^5 + 2*x^6 + 3*x^7 + 6*x^8 + 10*x^9 + 21*x^10 +...

%e g(g(x)) = x^4 + 2*x^5 + 4*x^6 + 7*x^7 + 14*x^8 + 26*x^9 + 52*x^10 +...

%e g(g(g(x))) = x^8 + 4*x^9 + 12*x^10 + 30*x^11 + 73*x^12 + 170*x^13 +...

%e g(g(g(g(x)))) = x^16 + 8*x^17 + 40*x^18 + 156*x^19 + 530*x^20 +...

%e where A(x) = x + g(x) + g(g(x)) + g(g(g(x))) + g(g(g(g(x)))) +...

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

%o for(n=1,45,print1(a(n),", "))

%K nonn

%O 1,4

%A _Paul D. Hanna_, May 11 2012