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G.f. satisfies: A(x) = (1 - x^3*A(x)^6) / (1 - x*A(x)^2)^2.
1

%I #8 May 31 2015 15:09:43

%S 1,2,11,79,647,5727,53367,515802,5123303,51977485,536320688,

%T 5610909773,59379328267,634538481389,6837466955193,74210071037031,

%U 810527496757335,8901979424068377,98253966680382102,1089260346498608721,12123804391067414676,135427509933882292680,1517725698030921469890

%N G.f. satisfies: A(x) = (1 - x^3*A(x)^6) / (1 - x*A(x)^2)^2.

%C Self-convolution yields A253255.

%F G.f. A(x) satisfies:

%F (1) A(x) = exp( Sum_{n>=1} A168595(n)/2 * x^n/n ), where A168595(n) = Sum_{k=0..2*n} binomial(n,k)*trinomial(n,k).

%F (2) A(x) = sqrt( (1/x)*Series_Reversion( x*(1-x)^4/(1-x^3)^2 ) ).

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

%e G.f.: A(x) = 1 + 2*x + 11*x^2 + 79*x^3 + 647*x^4 + 5727*x^5 + 53367*x^6 +...

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

%e The logarithm begins:

%e log(A(x)) = 2*x + 18*x^2 + 179*x^3 + 1874*x^4 + 20202*x^5 + 221943*x^6 + 2470827*x^7/7 +...+ A168595(n)/2*x^n/n +...

%o (PARI) {a(n) = local(A=1); A = sqrt( (1/x)*serreverse( x*(1-x)^4/(1-x^3)^2 +x^2*O(x^n))); polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {A168595(n) = sum(k=0, 2*n, binomial(2*n, k) * polcoeff((1+x+x^2)^n, k) )}

%o {a(n) = local(A=1); A = exp( sum(k=1,n+1, A168595(k)/2 * x^k/k) +x*O(x^n)); polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A168595, A106228, A064641, A253255.

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 31 2015

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