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A168598 G.f.: exp( Sum_{n>=1} A002426(n)^2*x^n/n ), where A002426(n) is the central trinomial coefficients.

%I A168598 #8 Mar 16 2021 08:28:14
%S A168598 1,1,5,21,119,703,4515,30227,210274,1503930,11008198,82099262,
%T A168598 622013122,4775754930,37089503826,290914775618,2301706690657,
%U A168598 18351027768401,147308337621061,1189704370416949,9661185599013209,78844977025403657
%N A168598 G.f.: exp( Sum_{n>=1} A002426(n)^2*x^n/n ), where A002426(n) is the central trinomial coefficients.
%C A168598 Compare to: exp( Sum_{n>=1} A002426(n)*x^n/n ) = g.f. of the Motzkin numbers (A001006).
%H A168598 G. C. Greubel, <a href="/A168598/b168598.txt">Table of n, a(n) for n = 0..250</a>
%e A168598 G.f.: A(x) = 1 + x + 5*x^2 + 21*x^3 + 119*x^4 + 703*x^5 +...
%e A168598 log(A(x)) = x + 9*x^2/2 + 49*x^3/3 + 361*x^4/4 + 2601*x^5/5 + 19881*x^6/6 +...+ A002426(n)^2*x^n/n +...
%t A168598 A002426[n_]:= GegenbauerC[n, -n, -1/2];
%t A168598 With[{m=30}, CoefficientList[Series[Exp[Sum[A002426[j]^2*x^j/j, {j, m+2}]], {x, 0, m}], x]] (* _G. C. Greubel_, Mar 16 2021 *)
%o A168598 (PARI) {a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,polcoeff((1+x+x^2)^m,m)^2*x^m/m)+x*O(x^n)),n))}
%o A168598 (Magma)
%o A168598 m:=30;
%o A168598 A002426:= func< n | (&+[ Binomial(n, k)*Binomial(k, n-k): k in [0..n]]) >;
%o A168598 R<x>:=PowerSeriesRing(Rationals(), m);
%o A168598 Coefficients(R!( Exp( (&+[A002426(j)^2*x^j/j: j in [1..m+2]]) ) )); // _G. C. Greubel_, Mar 16 2021
%o A168598 (Sage)
%o A168598 m=30
%o A168598 def A002426(n): return sum( binomial(n, k)*binomial(k, n-k) for k in (0..n) )
%o A168598 def A168598_list(prec):
%o A168598     P.<x> = PowerSeriesRing(QQ, prec)
%o A168598     return P( exp( sum( A002426(j)^2*x^j/j for j in [1..m+2])) ).list()
%o A168598 A168598_list(m) # _G. C. Greubel_, Mar 16 2021
%Y A168598 Cf. A001006, A002426, A168597, A168599.
%K A168598 nonn
%O A168598 0,3
%A A168598 _Paul D. Hanna_, Dec 01 2009