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A200377 G.f.: A(x) = exp( Sum_{n>=1} (Sum_{k=0..2*n} A027907(n,k)^2 * x^k / A(x)^k) * x^n/n ).

%I A200377 #10 Jul 06 2025 21:38:36
%S A200377 1,1,2,4,7,11,19,34,61,106,181,311,543,955,1668,2885,4980,8650,15114,
%T A200377 26391,45845,79385,137718,239866,418338,727926,1263097,2191463,
%U A200377 3810775,6638258,11556361,20078960,34855400,60567092,105405431,183483906,319039355,554158992,962743619,1674359119,2913758685,5068194691
%N A200377 G.f.: A(x) = exp( Sum_{n>=1} (Sum_{k=0..2*n} A027907(n,k)^2 * x^k / A(x)^k) * x^n/n ).
%e A200377 G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 7*x^4 + 11*x^5 + 19*x^6 + 34*x^7 +...
%e A200377 Let A = g.f. A(x), then the logarithm of the g.f. equals the series:
%e A200377 log(A(x)) = (1 + x/A + x^2/A^2)*x +
%e A200377 (1 + 2^2*x/A + 3^2*x^2/A^2 + 2^2*x^3/A^3 + x^4/A^4)*x^2/2 +
%e A200377 (1 + 3^2*x/A + 6^2*x^2/A^2 + 7^2*x^3/A^3 + 6^2*x^4/A^4 + 3^2*x^5/A^5 + x^6/A^6)*x^3/3 +
%e A200377 (1 + 4^2*x/A + 10^2*x^2/A^2 + 16^2*x^3/A^3 + 19^2*x^4/A^4 + 16^2*x^5/A^5 + 10^2*x^6/A^6 + 4^2*x^7/A^7 + x^8/A^8)*x^4/4 +
%e A200377 (1 + 5^2*x/A + 15^2*x^2/A^2 + 30^2*x^3/A^3 + 45^2*x^4/A^4 + 51^2*x^5/A^5 + 45^2*x^6/A^6 + 30^2*x^7/A^7 + 15^2*x^8/A^8 + 5^2*x^9/A^9 + x^10/A^10)*x^5/5 +...
%e A200377 which involves the squares of the trinomial coefficients A027907(n,k).
%o A200377 (PARI) /* G.f. A(x) using the squares of the trinomial coefficients */
%o A200377 {A027907(n, k)=polcoeff((1+x+x^2)^n, k)}
%o A200377 {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, 2*m, A027907(m, k)^2 *x^k/(A+x*O(x^n))^k) *x^m/m))); polcoeff(A, n)}
%Y A200377 Cf. A199248, A186236, A199257, A168592, A027907.
%K A200377 nonn
%O A200377 0,3
%A A200377 _Paul D. Hanna_, Nov 17 2011