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%I #12 Feb 20 2021 14:46:07
%S 1,1,1,2,6,20,70,255,961,3726,14797,59986,247606,1038632,4420837,
%T 19071954,83321966,368400431,1647706426,7452622503,34082926816,
%U 157595263361,736806253045,3483636843142,16660303710511,80618576499123,394863246977469,1958369414771028
%N G.f. A(x) satisfies: A(x) = (1 - x*A(x)) * Sum_{n>=0} x^n / (1 - x*A(x)^(n+1)).
%C The g.f. of this sequence is motivated by the following identity:
%C Sum_{n>=0} p^n/(1 - q*r^n) = Sum_{n>=0} q^n/(1 - p*r^n) = Sum_{n>=0} p^n*q^n*r^(n^2)*(1 - p*q*r^(2*n))/((1 - p*r^n)*(1-q*r^n)) ;
%C here, p = x, q = x*A(x), and r = A(x).
%F G.f. A(x) satisfies:
%F (1) A(x) = (1 - x*A(x)) * Sum_{n>=0} x^n / (1 - x*A(x)^(n+1)).
%F (2) A(x) = (1 - x*A(x)) * Sum_{n>=0} x^n*A(x)^n / (1 - x*A(x)^n).
%F (3) A(x) = (1 - x*A(x)) * Sum_{n>=0} x^(2*n) * A(x)^(n^2+n) * (1 - x^2*A(x)^(2*n+1)) / ((1 - x*A(x)^(n+1))*(1 - x*A(x)^n)). - _Paul D. Hanna_, Feb 20 2021
%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 70*x^6 + 255*x^7 + 961*x^8 + 3726*x^9 + 14797*x^10 + 59986*x^11 + 247606*x^12 + ...
%e where
%e A(x)/(1 - x*A(x)) = 1/(1 - x*A(x)) + x/(1 - x*A(x)^2) + x^2/(1 - x*A(x)^3) + x^3/(1 - x*A(x)^4) + x^4/(1 - x*A(x)^5) + ...
%e also
%e A(x)/(1 - x*A(x)) = 1/(1-x) + x*A(x)/(1 - x*A(x)) + x^2*A(x)^2/(1 - x*A(x)^2) + x^3*A(x)^3/(1 - x*A(x)^3) + x^4*A(x)^4/(1 - x*A(x)^4) + ...
%o (PARI) {a(n) = my(A=1); for(i=1,n, A = (1 - x*A) * sum(m=0,n, x^m / (1 - x*A^(m+1) +x*O(x^n)) ) ); polcoeff(H=A,n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n) = my(A=1); for(i=1,n, A = (1 - x*A) * sum(m=0,n, x^m*A^m / (1 - x*A^m +x*O(x^n)) ) ); polcoeff(H=A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A340361.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Jan 25 2021
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