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%I A341382 #6 Feb 11 2021 10:31:28
%S A341382 1,1,2,6,22,91,407,1921,9429,47683,246901,1303346,6992087,38031159,
%T A341382 209348857,1164616227,6540112446,37040976542,211423703225,
%U A341382 1215450693258,7034282005208,40966313765380,240003678300088,1414101405300096
%N A341382 G.f. A(x) satisfies: A(x) = P(x)/Q(x) where P(x) = Sum_{n>=0} (n+1)*x^n*A(x)^n/(1 - x*A(x)^(n+1)) and Q(x) = Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(n+1)).
%F A341382 G.f. A(x) satisfies:
%F A341382 (1) A(x) = P(x)/Q(x) where
%F A341382 P(x) = Sum_{n>=0} (n+1) * x^n * A(x)^n / (1 - x*A(x)^(n+1)) and
%F A341382 Q(x) = Sum_{n>=0} x^n * A(x)^n / (1 - x*A(x)^(n+1)).
%F A341382 (2) A(x) = P(x)/Q(x) where
%F A341382 P(x) = Sum_{n>=0} x^n * A(x)^n / (1 - x*A(x)^(n+1))^2 and
%F A341382 Q(x) = Sum_{n>=0} x^n * A(x)^n / (1 - x*A(x)^(n+1)).
%e A341382 G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 22*x^4 + 91*x^5 + 407*x^6 + 1921*x^7 + 9429*x^8 + 47683*x^9 + 246901*x^10 + 1303346*x^11 + 6992087*x^12 + ...
%e A341382 such that A(x) = P(x)/Q(x) where
%e A341382 P(x) = 1/(1-x*A(x)) + 2*x*A(x)/(1 - x*A(x)^2) + 3*x^2*A(x)^2/(1 - x*A(x)^3) + 4*x^3*A(x)^3/(1 - x*A(x)^4) + 5*x^4*A(x)^4/(1 - x*A(x)^5) + ...
%e A341382 Q(x) = 1/(1-x*A(x)) + x*A(x)/(1 - x*A(x)^2) + x^2*A(x)^2/(1 - x*A(x)^3) + x^3*A(x)^3/(1 - x*A(x)^4) + x^4*A(x)^4/(1 - x*A(x)^5) + ...
%e A341382 also
%e A341382 P(x) = 1/(1-x*A(x))^2 + x*A(x)/(1 - x*A(x)^2)^2 + x^2*A(x)^2/(1 - x*A(x)^3)^2 + x^3*A(x)^3/(1 - x*A(x)^4)^2 + x^4*A(x)^4/(1 - x*A(x)^5)^2 + ...
%e A341382 explicitly,
%e A341382 P(x) = 1 + 3*x + 9*x^2 + 30*x^3 + 111*x^4 + 448*x^5 + 1937*x^6 + 8837*x^7 + 42046*x^8 + 206821*x^9 + 1044977*x^10 + 5397263*x^11 + ...
%e A341382 Q(x) = 1 + 2*x + 5*x^2 + 15*x^3 + 52*x^4 + 201*x^5 + 843*x^6 + 3760*x^7 + 17579*x^8 + 85259*x^9 + 425772*x^10 + 2177369*x^11 + ...
%o A341382 (PARI) {a(n) = my(A=1+x+x*O(x^n), P=1, Q=1); for(i=0, n,
%o A341382 P = sum(m=0, n, (m+1)*x^m*A^m/(1 - x*A^(m+1) + x*O(x^n)) );
%o A341382 Q = sum(m=0, n, x^m*A^m/(1 - x*A^(m+1) + x*O(x^n)) );
%o A341382 A = P/Q); polcoeff(H=A, n)}
%o A341382 for(n=0, 30, print1(a(n), ", "))
%o A341382 (PARI) {a(n) = my(A=1+x+x*O(x^n), P=1, Q=1); for(i=0, n,
%o A341382 P = sum(m=0, n, x^m*A^m/(1 - x*A^(m+1) + x*O(x^n))^2 );
%o A341382 Q = sum(m=0, n, x^m*A^m/(1 - x*A^(m+1) + x*O(x^n)) );
%o A341382 A = P/Q); polcoeff(H=A, n)}
%o A341382 for(n=0, 30, print1(a(n), ", "))
%Y A341382 Cf. A341342.
%K A341382 nonn
%O A341382 0,3
%A A341382 _Paul D. Hanna_, Feb 10 2021

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