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A218299 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..2*n} A084606(n,k)^2 * x^k * A(x)^k ), where A084606(n,k) = [x^k] (1 + 2*x + 2*x^2)^n.

%I A218299 #6 Jan 25 2025 18:59:23
%S A218299 1,1,5,21,109,573,3209,18425,108649,652425,3979805,24583853,153488501,
%T A218299 966993893,6139832385,39249227569,252400089361,1631676380497,
%U A218299 10597809743477,69123464993925,452567027633853,2973269053045197,19595030047168569,129509530910221737
%N A218299 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..2*n} A084606(n,k)^2 * x^k * A(x)^k  ), where A084606(n,k) = [x^k] (1 + 2*x + 2*x^2)^n.
%F A218299 G.f. satisfies:
%F A218299 (1) A(x) = (1 + x*A(x))*(1 + 4*x^3*A(x)^3)*(1 + 4*x^4*A(x)^4)/(1 - 2*x^2*A(x)^2)^2.
%F A218299 (3) A(x) = (1/x)*Series_Reversion( x*(1 - 2*x^2)^2 / ((1 + x)*(1 + 4*x^3)*(1 + 4*x^4)) ).
%e A218299 G.f.: A(x) = 1 + x + 5*x^2 + 21*x^3 + 109*x^4 + 573*x^5 + 3209*x^6 +...
%e A218299 Let A = g.f. A(x), then the logarithm of the g.f. equals the series:
%e A218299 log(A(x)) = (1 + 2^2*x*A + 2^2*x^2*A^2)*x +
%e A218299 (1 + 4^2*x*A + 8^2*x^2*A^2 + 8^2*x^3*A^3 + 4^2*x^4*A^4)*x^2/2 +
%e A218299 (1 + 6^2*x*A + 18^2*x^2*A^2 + 32^2*x^3*A^3 + 36^2*x^4*A^4 + 24^2*x^5*A^5 + 8^2*x^6*A^6)*x^3/3 +
%e A218299 (1 + 8^2*x*A + 32^2*x^2*A^2 + 80^2*x^3*A^3 + 136^2*x^4*A^4 + 160^2*x^5*A^5 + 128^2*x^6*A^6 + 64^2*x^7*A^7 + 16^2*x^8*A^8)*x^4/4 +...
%e A218299 which involves the squares of the trinomial coefficients A084606(n,k):
%e A218299 1;
%e A218299 1, 2, 2;
%e A218299 1, 4, 8, 8, 4;
%e A218299 1, 6, 18, 32, 36, 24, 8;
%e A218299 1, 8, 32, 80, 136, 160, 128, 64, 16;
%e A218299 1, 10, 50, 160, 360, 592, 720, 640, 400, 160, 32; ...
%o A218299 (PARI) /* G.f. A(x) using the squares of the trinomial coefficients A084606: */
%o A218299 {A084606(n, k)=polcoeff((1 + 2*x + 2*x^2)^n, k)}
%o A218299 {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, 2*m, A084606(m, k)^2*x^k*(A+x*O(x^n))^k)*x^m/m))); polcoeff(A, n)}
%o A218299 for(n=0,20,print1(a(n),", "))
%o A218299 (PARI) {a(n)=local(A=serreverse( x*(1 - 2*x^2)^2 / ((1 + x)*(1 + 4*x^3)*(1 + 4*x^4 +x*O(x^n))))/x);polcoeff(A,n)}
%o A218299 for(n=0,20,print1(a(n),", "))
%Y A218299 Cf. A218298, A200475, A084606.
%K A218299 nonn
%O A218299 0,3
%A A218299 _Paul D. Hanna_, Oct 28 2012