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A375436 Expansion of g.f. A(x) satisfying A(x) = (1 + 2*x*A(x)) * (1 + 3*x*A(x)^2).

%I A375436 #8 Sep 08 2024 19:48:26
%S A375436 1,5,46,533,6922,96338,1404796,21184229,327659314,5169425894,
%T A375436 82866843652,1345864066658,22098946620580,366245357320196,
%U A375436 6118363978530424,102921394554326021,1741855452305095618,29637960953559091934,506708801920060974388,8700147627314354759030,149957787462657877848556
%N A375436 Expansion of g.f. A(x) satisfying A(x) = (1 + 2*x*A(x)) * (1 + 3*x*A(x)^2).
%C A375436 In general, if G(x) = (1 + p*x*G(x)) * (1 + q*x*G(x)^2) for fixed p and q, then
%C A375436 (C.1) G(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * p^(n-k) * q^k * G(x)^k ).
%C A375436 (C.2) G(x) = (1/x) * Series_Reversion( x/(1 + p*x) - q*x^2 ).
%C A375436 (C.3) x = (sqrt((p - q*y)^2 + 4*p*q*y^2) - (p + q*y))/(2*p*q*y^2), where y = G(x).
%H A375436 Paul D. Hanna, <a href="/A375436/b375436.txt">Table of n, a(n) for n = 0..400</a>
%F A375436 G.f. A(x) = Sum{n>=0} a(n)*x^n satisfies the following formulas.
%F A375436 (1) A(x) = (1 + 2*x*A(x)) * (1 + 3*x*A(x)^2).
%F A375436 (2) A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * 2^(n-k) * 3^k * A(x)^k ).
%F A375436 (3) A(x) = (1/x) * Series_Reversion( x*(1 - 3*x - 6*x^2)/(1 + 2*x) ).
%F A375436 (4) A(x) = Sum_{n>=0} A154964(n+1) * x^n * A(x)^n, where g.f. of A154964 = (1 - 2*x - 4*x^2)/(1 - 3*x - 6*x^2).
%F A375436 (5) x = (sqrt(33*A(x)^2 - 12*A(x) + 4) - (2 + 3*A(x)))/(12*A(x)^2).
%F A375436 a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(n+k+1,k) * binomial(n+k+1,n-k) / (n+k+1). - _Seiichi Manyama_, Sep 08 2024
%e A375436 G.f. A(x) = 1 + 5*x + 46*x^2 + 533*x^3 + 6922*x^4 + 96338*x^5 + 1404796*x^6 + 21184229*x^7 + 327659314*x^8 + 5169425894*x^9 + 82866843652*x^10 + ...
%e A375436 where A(x) = (1 + 2*x*A(x)) * (1 + 3*x*A(x)^2).
%e A375436 RELATED SERIES.
%e A375436 Let B(x) = A(x/B(x)) and B(x*A(x)) = A(x), then
%e A375436 B(x) = 1 + 5*x + 21*x^2 + 93*x^3 + 405*x^4 + 1773*x^5 + 7749*x^6 + 33885*x^7 + ... + A154964(n+1)*x^n + ...
%e A375436 where B(x) = (1 + 2*x)/(1 - 3*x - 6*x^2).
%o A375436 (PARI) {a(n) = my(A=1+x); for(i=1, n, A=(1 + 2*x*A)*(1 + 3*x*(A+x*O(x^n))^2)); polcoef(A, n)}
%o A375436 for(n=0, 20, print1(a(n), ", "))
%o A375436 (PARI) {a(n)=polcoef( (1/x)*serreverse( x*(1 - 3*x - 6*x^2)/(1 + 2*x +x*O(x^n))), n)}
%o A375436 for(n=0, 20, print1(a(n), ", "))
%o A375436 (PARI) {a(n) = my(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2 * 2^(m-j) * 3^j * A^j)*x^m/m))); polcoef(A, n)}
%o A375436 for(n=0, 20, print1(a(n), ", "))
%Y A375436 Cf. A154964, A216314, A215661, A375434, A375435, A375437.
%K A375436 nonn
%O A375436 0,2
%A A375436 _Paul D. Hanna_, Sep 07 2024