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G.f.: exp( Sum_{n>=1} A188202(n)*x^n/n ) where A188202(n) = [x^n] (1 + 2^n*x + x^2)^n.
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%I #4 Mar 30 2012 18:37:26

%S 1,2,11,206,17586,6878604,11551087875,80650796495414,

%T 2307974943300931286,268728588584911887188180,

%U 126776477973814964972206209838,241684409250478693507166916367088620

%N G.f.: exp( Sum_{n>=1} A188202(n)*x^n/n ) where A188202(n) = [x^n] (1 + 2^n*x + x^2)^n.

%C Compare to the g.f. M(x) of the Motzkin numbers (A001006):

%C M(x) = exp( Sum_{n>=1} A002426(n)*x^n/n) where A002426(n) = [x^n] (1+x+x^2)^n.

%e G.f.: A(x) = 1 + 2*x + 11*x^2 + 206*x^3 + 17586*x^4 + 6878604*x^5 +...

%e The l.g.f. of A188202 begins:

%e log(A(x)) = 2*x + 18*x^2/2 + 560*x^3/3 + 68614*x^4/4 + 34210752*x^5/5 +...

%e The coefficients of x^n in (1 + 2^n*x + x^2)^n begin:

%e n=1: [1, (2), 1];

%e n=2: [1, 8, (18), 8, 1];

%e n=3: [1, 24, 195, (560), 195, 24, 1];

%e n=4: [1, 64, 1540, 16576, (68614), 16576, 1540, 64, 1];

%e n=5: [1, 160, 10245, 328320, 5273610, (34210752), 5273610, ...]; ...

%e where the central coefficients form the logarithmic derivative, A188202.

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

%Y Cf. A188202 (log); variant: A001006.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 24 2011