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%I #16 Jul 30 2018 12:45:01
%S 1,1,2,5,15,50,182,707,2903,12479,55844,258860,1238588,6099054,
%T 30836886,159770751,846927495,4586883023,25351486346,142843162421,
%U 819783142271,4788268962584,28444114318056,171737405798836,1053285775758916,6558551535958516,41441942236323008
%N G.f. satisfies: A(x) = 1 + x*A(x) + x^2*A(x)^2 + x^3*A(x)*A'(x).
%H Paul D. Hanna, <a href="/A245311/b245311.txt">Table of n, a(n) for n = 0..500</a>
%F G.f. A(x) satisfies:
%F (1) A(x) = sqrt( Dx(A(x)) / (1 + x^2*Dx^2(A(x))) ), where Dx(F(x)) = d/dx x*F(x).
%F (2) A(x) = sqrt( (A(x) + x*A'(x)) / (1 + x^2*A(x) + 3*x^3*A'(x) + x^4*A''(x)) ).
%F (3) A(x) = sqrt( A'(x) / (1 + 2*x*A(x) + 4*x^2*A'(x) + x^3*A''(x)) ).
%F (4) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A245310.
%F (5) [x^n] (1-n + n*A(x)) * exp(-n*x*A(x)) = 0 for n>=1. - _Paul D. Hanna_, Jul 30 2018
%F a(n) = [x^n] G(x)^(n+1)/(n+1) for n>=0 where G(x) is the g.f. of A245310.
%F a(n) = [x^(n+2)] G(x)^(n+1)/(n+1)^2 for n>=0 where G(x) is the g.f. of A245310.
%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 50*x^5 + 182*x^6 + 707*x^7 +...
%e where A(x) = 1 / (1-x - x^2*A(x) - x^3*A'(x)).
%e Define Dx(A(x)) = d/dx x*A(x) = A(x) + x*A'(x):
%e Dx(A(x)) = 1 + 2*x + 6*x^2 + 20*x^3 + 75*x^4 + 300*x^5 + 1274*x^6 +...
%e so that Dx^2(A(x)) = d/dx x*(d/dx x*A(x)) = A(x) + 3*x*A'(x) + x^2*A''(x):
%e Dx^2(A(x)) = 1 + 4*x + 18*x^2 + 80*x^3 + 375*x^4 + 1800*x^5 + 8918*x^6 +...
%e then A(x)^2 = Dx(A(x)) / (1 + x^2*Dx^2(A(x))):
%e A(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 44*x^4 + 150*x^5 + 549*x^6 + 2128*x^7 + 8673*x^8 + 36912*x^9 + 163288*x^10 +...
%e RELATED SERIES.
%e The g.f. of A245310 begins:
%e G(x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 12*x^6 + 34*x^7 + 120*x^8 +...
%e where A(x) = G(x*A(x)) where G(x) = A(x/G(x)) and
%e a(n) = [x^n] G(x)^(n+1)/(n+1) = [x^(n+2)] G(x)^(n+1)/(n+1)^2 for n>=0.
%o (PARI) /* From A(x) = 1 + x*A(x) + x^2*A(x)^2 + x^3*A(x)*A'(x): */
%o {a(n)=local(A=1+x);for(i=1,n,A = (1 + x^2*A^2 + x^3*A*A')/(1-x +x*O(x^n)));
%o polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n)=local(A=1+x+2*x^2+sum(k=3,n-1,a(k)*x^k) +x^2*O(x^n));
%o if(n<3,polcoeff(A,n),-polcoeff( sqrt( deriv(x*A) / (1 + x^2*deriv(x*deriv(x*A)) +x^2*O(x^n)) ), n)/((n-1)/2))}
%o for(n=0,25,print1(a(n),", "))
%o (PARI) /* From A(x) = G(x*A(x)) where G(x) is the g.f. of A245310: */
%o {a(n)=local(G=1+x+x^2);for(m=2,n,
%o G=G-x^(m+1)*(polcoeff(G^m+O(x^(m+2)),m+1)/m - polcoeff(G^m+O(x^m),m-1))+O(x^(n+3)));
%o polcoeff(1/x*serreverse(x/G+O(x^(n+3))),n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A245310, A245313.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jul 17 2014
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