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%I #11 Oct 25 2016 21:35:09
%S 1,2,9,56,420,3572,33328,334354,3559310,39838760,465743720,5658983108,
%T 71191948512,924554859776,12365546196641,169995491295312,
%U 2398380272232272,34680290150700800,513390937937217088,7773229533145403728,120277760289804227632,1900583166564027019136,30649888151334972466392,504153517331248726221392,8454018409655883681321232,144451967918022160558965408
%N G.f. A(x) satisfies: A( x - A(x) ) = x^2.
%C An unsigned version of A138740 (apart from initial term).
%C Self-convolution of A213591.
%H Paul D. Hanna, <a href="/A276370/b276370.txt">Table of n, a(n) for n = 2..302</a>
%F G.f. satisfies:
%F (1) A(x) = ( x + A( sqrt(A(x)) ) )^2.
%F (2) A(x) = ( x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n / n! )^2.
%F (3) A(x) = x^2 * exp( 2*Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n/x / n! ).
%F (4) A(x) = x^2 * G( A(x)/x )^2 where G(x) = 1 + x*G( 1 - 1/G(x) )^2 is the g.f. of A212411.
%F (5) A(x) = x^2 * F( sqrt(A(x)) )^2 where F(x) = 1 - (x^2/F(x)) / F( x^2/F(x) ) is the g.f. of A213628.
%e G.f.: A(x) = x^2 + 2*x^3 + 9*x^4 + 56*x^5 + 420*x^6 + 3572*x^7 + 33328*x^8 + 334354*x^9 + 3559310*x^10 + 39838760*x^11 + 465743720*x^12 +...
%e such that A( x - A(x) ) = x^2.
%o (PARI) {a(n) = local(A=x^2); for(i=1, n, A = serreverse(x - A +x*O(x^n))^2); polcoeff(A, n)}
%o for(n=2,30,print1(a(n),", "))
%o (PARI) {Dx(n, F) = local(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n) = local(A=x^2 +x*O(x^n)); for(i=1, n, A = (x + sum(m=1, n, Dx(m-1, A^m)/m!) +x*O(x^n))^2); polcoeff(A, n)}
%o for(n=2,30,print1(a(n),", "))
%Y Cf. A138740, A213591, A212411, A213628.
%K nonn
%O 2,2
%A _Paul D. Hanna_, Sep 09 2016
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