%I #12 Sep 08 2024 19:48:58

%S 1,5,41,427,4997,62697,824361,11210331,156371609,2224976461,

%T 32167995497,471208730027,6978452945485,104313403711649,

%U 1571764793999769,23847629857934859,364033580432140593,5586881305151655381,86153520326218040553,1334246446733337499755,20743139707001572645461

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

%C In general, if G(x) = (1 + p*x*G(x)) * (1 + q*x*G(x)^2) for fixed p and q, then

%C (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 (C.2) G(x) = (1/x) * Series_Reversion( x/(1 + p*x) - q*x^2 ).

%C (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 Paul D. Hanna, <a href="/A375437/b375437.txt">Table of n, a(n) for n = 0..400</a>

%F G.f. A(x) = Sum{n>=0} a(n)*x^n satisfies the following formulas.

%F (1) A(x) = (1 + 3*x*A(x)) * (1 + 2*x*A(x)^2).

%F (2) A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * 3^(n-k) * 2^k * A(x)^k ).

%F (3) A(x) = (1/x) * Series_Reversion( x*(1 - 2*x - 6*x^2)/(1 + 3*x) ).

%F (4) A(x) = Sum_{n>=0} A307469(n) * x^n * A(x)^n, where g.f. of A307469 = (1 + 3*x)/(1 - 3*x - 6*x^2).

%F (5) x = (sqrt(28*A(x)^2 - 12*A(x) + 9) - (3 + 2*A(x)))/(12*A(x)^2).

%F a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(n+k+1,k) * binomial(n+k+1,n-k) / (n+k+1). - _Seiichi Manyama_, Sep 08 2024

%e G.f. A(x) = 1 + 5*x + 41*x^2 + 427*x^3 + 4997*x^4 + 62697*x^5 + 824361*x^6 + 11210331*x^7 + 156371609*x^8 + 2224976461*x^9 + 32167995497*x^10 + ...

%e where A(x) = (1 + 3*x*A(x)) * (1 + 2*x*A(x)^2).

%e RELATED SERIES.

%e Let B(x) = A(x/B(x)) and B(x*A(x)) = A(x), then

%e B(x) = 1 + 5*x + 16*x^2 + 62*x^3 + 220*x^4 + 812*x^5 + 2944*x^6 + 10760*x^7 + ... + A307469(n)*x^n + ...

%e where B(x) = (1 + 3*x)/(1 - 3*x - 6*x^2).

%o (PARI) {a(n) = my(A=1+x); for(i=1, n, A=(1 + 3*x*A)*(1 + 2*x*(A+x*O(x^n))^2)); polcoef(A, n)}

%o for(n=0, 20, print1(a(n), ", "))

%o (PARI) {a(n)=polcoef( (1/x)*serreverse( x*(1 - 2*x - 6*x^2)/(1 + 3*x +x*O(x^n))), n)}

%o for(n=0, 20, print1(a(n), ", "))

%o (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 * 3^(m-j) * 2^j * A^j)*x^m/m))); polcoef(A, n)}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A307469, A216314, A215661, A375434, A375435, A375436.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Sep 07 2024