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%I #5 Aug 04 2012 01:26:06
%S 1,8,108,1604,25000,400680,6539932,108124704,1804745052,30345973020,
%T 513246523504,8722051475280,148808353519644,2547317293116512,
%U 43729600850054640,752553083643333008,12978731348733064680,224258766666165861192
%N G.f.: Sum_{n>=0} d^n/dx^n (2*x + x^2)^(2*n) / n!.
%C Compare to: Sum_{n>=0} d^n/dx^n x^(2*n)/n! = 1/sqrt(1-4*x).
%F G.f.: A(x) = d/dx G(x), where G(x) = x + (2*G(x) + G(x)^2)^2.
%e G.f.: A(x) = 1 + 2*x + 12*x^2 + 64*x^3 + 370*x^4 + 2184*x^5 + 13132*x^6 +...
%e such that, by definition:
%e A(x) = 1 + d/dx (2*x+x^2)^2 + d^2/dx^2 (2*x+x^2)^4/2! + d^3/dx^3 (2*x+x^2)^6/3! + d^4/dx^4 (2*x+x^2)^8/4! + d^5/dx^5 (2*x+x^2)^10/5! +...
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n)=local(A=x); A=1+sum(m=1, n, Dx(m, (2*x+x^2+x*O(x^n))^(2*m)/m!)); polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A215128.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Aug 04 2012
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