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A168599 G.f.: exp( Sum_{n>=1} A002426(n)^n * x^n/n ), where A002426(n) is the central trinomial coefficients.

%I A168599 #8 Mar 16 2021 08:28:21
%S A168599 1,1,5,119,32707,69038213,1309743837515,206848589180297555,
%T A168599 281897548265847120670891,3287603007740009094151486257065,
%U A168599 330891681467139744269091005122077348971
%N A168599 G.f.: exp( Sum_{n>=1} A002426(n)^n * x^n/n ), where A002426(n) is the central trinomial coefficients.
%C A168599 Compare to: exp( Sum_{n>=1} A002426(n)*x^n/n ) = g.f. of the Motzkin numbers (A001006).
%H A168599 G. C. Greubel, <a href="/A168599/b168599.txt">Table of n, a(n) for n = 0..45</a>
%e A168599 G.f.: A(x) = 1 + x + 5*x^2 + 119*x^3 + 32707*x^4 +...
%e A168599 log(A(x)) = x + 9*x^2/2 + 343*x^3/3 + 130321*x^4/4 +...+ A002426(n)^n*x^n/n +...
%p A168599 m:=30;
%p A168599 A002426:= n-> add( binomial(n, k)*binomial(k, n-k), k=0..n );
%p A168599 S := series( exp(add(A002426(j)^j*x^j/j, j = 1..m+2)), x, m+1);
%p A168599 seq(coeff(S, x, j), j = 0..m); # _G. C. Greubel_, Mar 16 2021
%t A168599 A002426[n_] := GegenbauerC[n, -n, -1/2];
%t A168599 With[{m=30}, CoefficientList[Series[Exp[Sum[A002426[j]^j*x^j/j, {j, m+2}]], {x, 0, m}], x]] (* _G. C. Greubel_, Mar 16 2021 *)
%o A168599 (PARI) {a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,polcoeff((1+x+x^2)^m,m)^m*x^m/m)+x*O(x^n)),n))}
%o A168599 (Magma)
%o A168599 m:=30;
%o A168599 A002426:= func< n | (&+[ Binomial(n, k)*Binomial(k, n-k): k in [0..n]]) >;
%o A168599 R<x>:=PowerSeriesRing(Rationals(), m);
%o A168599 Coefficients(R!( Exp( (&+[A002426(j)^j*x^j/j: j in [1..m+2]]) ) )); // _G. C. Greubel_, Mar 16 2021
%o A168599 (Sage)
%o A168599 m=30
%o A168599 def A002426(n): return sum( binomial(n, k)*binomial(k, n-k) for k in (0..n) )
%o A168599 def A168598_list(prec):
%o A168599     P.<x> = PowerSeriesRing(QQ, prec)
%o A168599     return P( exp( sum( A002426(j)^j*x^j/j for j in [1..m+2])) ).list()
%o A168599 A168598_list(m) # _G. C. Greubel_, Mar 16 2021
%Y A168599 Cf. A001006, A002426, A168598, A225328.
%K A168599 nonn
%O A168599 0,3
%A A168599 _Paul D. Hanna_, Dec 01 2009