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A186236 G.f.: exp( Sum_{n>=0} [ Sum_{k=0..2*n} A027907(n,k)^2 * x^k ]* x^n/n ), where A027907 is the triangle of trinomial coefficients.

%I A186236 #14 Mar 30 2012 18:37:26
%S A186236 1,1,2,5,13,34,93,262,753,2198,6502,19449,58724,178739,547836,1689407,
%T A186236 5237939,16318137,51056027,160363129,505456920,1598263936,5068483189,
%U A186236 16116397411,51371962474,164123564499,525447953073,1685534207788,5416719384326,17437073203711
%N A186236 G.f.: exp( Sum_{n>=0} [ Sum_{k=0..2*n} A027907(n,k)^2 * x^k ]* x^n/n ), where A027907 is the triangle of trinomial coefficients.
%C A186236 Trinomial coefficients satisfy: Sum_{k=0..2*n} A027907(n,k)*x^k = (1+x+x^2)^n.
%e A186236 G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 34*x^5 + 93*x^6 +...
%e A186236 The logarithm begins:
%e A186236 log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 31*x^4/4 + 91*x^5/5 + 282*x^6/6 + 890*x^7/7 + 2831*x^8/8 + 9055*x^9/9 + 29133*x^10/10 +...
%e A186236 which equals the sum of the series:
%e A186236 log(A(x)) = (1 + x + x^2)*x
%e A186236 + (1 + 2^2*x + 3^2*x^2 + 2^2*x^3 + x^4)*x^2/2
%e A186236 + (1 + 3^2*x + 6^2*x^2 + 7^2*x^3 + 6^2*x^4 + 3*x^5 + x^6)*x^3/3
%e A186236 + (1 + 4^2*x + 10^2*x^2 + 16^2*x^3 + 19^2*x^4 + 16^2*x^5 + 10^2*x^6 + 4^2*x^7 + x^8)*x^4/4
%e A186236 + (1 + 5^2*x + 15^2*x^2 + 30^2*x^3 + 45^2*x^4 + 51^2*x^5 + 45^2*x^6 + 30^2*x^7 + 15^2*x^8 + 5^2*x^9 + x^10)*x^5/5 +...
%o A186236 (PARI) {A027907(n,k)=polcoeff((1+x+x^2)^n, k)}
%o A186236 {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, 2*m, A027907(m,k)^2 *x^k) *x^m/m)+x*O(x^n)), n)}
%Y A186236 Cf. A180718 (variant).
%K A186236 nonn
%O A186236 0,3
%A A186236 _Paul D. Hanna_, Oct 19 2011