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A188203 G.f.: exp( Sum_{n>=1} A188202(n)*x^n/n ) where A188202(n) = [x^n] (1 + 2^n*x + x^2)^n.

%I A188203 #8 Dec 17 2024 17:06:59
%S A188203 1,2,11,206,17586,6878604,11551087875,80650796495414,
%T A188203 2307974943300931286,268728588584911887188180,
%U A188203 126776477973814964972206209838,241684409250478693507166916367088620
%N A188203 G.f.: exp( Sum_{n>=1} A188202(n)*x^n/n ) where A188202(n) = [x^n] (1 + 2^n*x + x^2)^n.
%C A188203 Compare to the g.f. M(x) of the Motzkin numbers (A001006):
%C A188203 M(x) = exp( Sum_{n>=1} A002426(n)*x^n/n) where A002426(n) = [x^n] (1+x+x^2)^n.
%e A188203 G.f.: A(x) = 1 + 2*x + 11*x^2 + 206*x^3 + 17586*x^4 + 6878604*x^5 +...
%e A188203 The l.g.f. of A188202 begins:
%e A188203 log(A(x)) = 2*x + 18*x^2/2 + 560*x^3/3 + 68614*x^4/4 + 34210752*x^5/5 +...
%e A188203 The coefficients of x^n in (1 + 2^n*x + x^2)^n begin:
%e A188203 n=1: [1, (2), 1];
%e A188203 n=2: [1, 8, (18), 8, 1];
%e A188203 n=3: [1, 24, 195, (560), 195, 24, 1];
%e A188203 n=4: [1, 64, 1540, 16576, (68614), 16576, 1540, 64, 1];
%e A188203 n=5: [1, 160, 10245, 328320, 5273610, (34210752), 5273610, ...]; ...
%e A188203 where the central coefficients form the logarithmic derivative, A188202.
%o A188203 (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 A188203 Cf. A188202 (log); variant: A001006.
%K A188203 nonn
%O A188203 0,2
%A A188203 _Paul D. Hanna_, Mar 24 2011