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%I #2 Mar 30 2012 18:37:17
%S 1,1,2,5,23,205,3833,148051,11761606,1909231503,632185554036,
%T 427306055229923,589583957310155426,1662328104286133851880,
%U 9585835617647933412333536,113145883593065457861894176545
%N G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n * A(n^2*x)^(1/n).
%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 23*x^4 + 205*x^5 + 3833*x^6 +...
%e A(4*x)^(1/2) = 1 + 2*x + 14*x^2 + 132*x^3 + 2582*x^4 + 97948*x^5 +...
%e A(9*x)^(1/3) = 1 + 3*x + 45*x^2 + 936*x^3 + 42255*x^4 +...
%e A(16*x)^(1/4) = 1 + 4*x + 104*x^2 + 3808*x^3 + 309856*x^4 +...
%e A(25*x)^(1/5) = 1 + 5*x + 200*x^2 + 11375*x^3 + 1458750*x^4 +...
%e A(36*x)^(1/6) = 1 + 6*x + 342*x^2 + 27900*x^3 + 5182758*x^4 +...
%e A(49*x)^(1/7) = 1 + 7*x + 539*x^2 + 59682*x^3 + 15155112*x^4 +...
%e A(64*x)^(1/8) = 1 + 8*x + 800*x^2 + 115456*x^3 + 38417920*x^4 +...
%e A(81*x)^(1/9) = 1 + 9*x + 1134*x^2 + 206793*x^3 + 87311601*x^4 +...
%e A(100*x)^(1/10) = 1 + 10*x + 1550*x^2 + 348500*x^3 + 182033750*x^4 +...
%e A(121*x)^(1/11) = 1 + 11*x + 2057*x^2 + 559020*x^3 + 353916893*x^4 +...
%e A(144*x)^(1/12) = 1 + 12*x + 2664*x^2 + 860832*x^3 + 649514592*x^4 +...
%e ...
%e Initial terms equal the antidiagonal sums of above coefficients:
%e a(1) = 1 ;
%e a(2) = 1 + 1 = 2 ;
%e a(3) = 1 + 2 + 2 = 5 ;
%e a(4) = 1 + 3 + 14 + 5 = 23 ;
%e a(5) = 1 + 4 + 45 + 132 + 23 = 205 ;
%e a(6) = 1 + 5 + 104 + 936 + 2582 + 205 = 3833 ;
%e a(7) = 1 + 6 + 200 + 3808 + 42255 + 97948 + 3833 = 148051 ; ...
%o (PARI) {a(n)=local(A=1+x);for(n=2,n, A=1 + sum(k=1,n,x^k*subst(A,x,k^2*x+x*O(x^n))^(1/k)));polcoeff(A,n)}
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 02 2009
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