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%I #7 Mar 30 2012 18:37:31
%S 1,1,2,5,19,160,3418,179705,19488053,4590422901,2738580784946,
%T 3583015072969210,9255051219746866753,56916338252385095986978,
%U 871826913772059843867743765,26753845554560439025697319191184,1695956186616651065722319776300825712
%N G.f.: exp( Sum_{n>=0} [ Sum_{k=0..2*n} A027907(n,k)^n * x^k ]* x^n/n ), where A027907 is the triangle of trinomial coefficients.
%C Trinomial coefficients satisfy: Sum_{k=0..2*n} A027907(n,k)*x^k = (1+x+x^2)^n.
%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 19*x^4 + 160*x^5 + 3418*x^6 +...
%e The logarithm begins:
%e log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 55*x^4/4 + 691*x^5/5 + 19440*x^6/6 + 1232750*x^7/7 + 154436735*x^8/8 + 41136723397*x^9/9 +...
%e which equals the sum of the series:
%e log(A(x)) = (1 + x + x^2)*x
%e + (1 + 2^2*x + 3^2*x^2 + 2^2*x^3 + x^4)*x^2/2
%e + (1 + 3^3*x + 6^3*x^2 + 7^3*x^3 + 6^3*x^4 + 3^3*x^5 + x^6)*x^3/3
%e + (1 + 4^4*x + 10^4*x^2 + 16^4*x^3 + 19^4*x^4 + 16^4*x^5 + 10^4*x^6 + 4^4*x^7 + x^8)*x^4/4
%e + (1 + 5^5*x + 15^5*x^2 + 30^5*x^3 + 45^5*x^4 + 51^5*x^5 + 45^5*x^6 + 30^5*x^7 + 15^5*x^8 + 5^5*x^9 + x^10)*x^5/5 +...
%o (PARI) {A027907(n, k)=polcoeff((1+x+x^2)^n, k)}
%o {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, 2*m, A027907(m, k)^m *x^k) *x^m/m)+x*O(x^n)), n)}
%Y Cf. A186236, A168590, A027907.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 22 2011
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