A168599 - OEIS

%I #8 Mar 16 2021 08:28:21

%S 1,1,5,119,32707,69038213,1309743837515,206848589180297555,

%T 281897548265847120670891,3287603007740009094151486257065,

%U 330891681467139744269091005122077348971

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

%C Compare to: exp( Sum_{n>=1} A002426(n)*x^n/n ) = g.f. of the Motzkin numbers (A001006).

%H G. C. Greubel, <a href="/A168599/b168599.txt">Table of n, a(n) for n = 0..45</a>

%e G.f.: A(x) = 1 + x + 5*x^2 + 119*x^3 + 32707*x^4 +...

%e log(A(x)) = x + 9*x^2/2 + 343*x^3/3 + 130321*x^4/4 +...+ A002426(n)^n*x^n/n +...

%p m:=30;

%p A002426:= n-> add( binomial(n, k)*binomial(k, n-k), k=0..n );

%p S := series( exp(add(A002426(j)^j*x^j/j, j = 1..m+2)), x, m+1);

%p seq(coeff(S, x, j), j = 0..m); # _G. C. Greubel_, Mar 16 2021

%t A002426[n_] := GegenbauerC[n, -n, -1/2];

%t 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 (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 (Magma)

%o m:=30;

%o A002426:= func< n | (&+[ Binomial(n, k)*Binomial(k, n-k): k in [0..n]]) >;

%o R<x>:=PowerSeriesRing(Rationals(), m);

%o Coefficients(R!( Exp( (&+[A002426(j)^j*x^j/j: j in [1..m+2]]) ) )); // _G. C. Greubel_, Mar 16 2021

%o (Sage)

%o m=30

%o def A002426(n): return sum( binomial(n, k)*binomial(k, n-k) for k in (0..n) )

%o def A168598_list(prec):

%o     P.<x> = PowerSeriesRing(QQ, prec)

%o     return P( exp( sum( A002426(j)^j*x^j/j for j in [1..m+2])) ).list()

%o A168598_list(m) # _G. C. Greubel_, Mar 16 2021

%Y Cf. A001006, A002426, A168598, A225328.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 01 2009