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

1, 1, 5, 21, 119, 703, 4515, 30227, 210274, 1503930, 11008198, 82099262, 622013122, 4775754930, 37089503826, 290914775618, 2301706690657, 18351027768401, 147308337621061, 1189704370416949, 9661185599013209, 78844977025403657

Offset: 0

Author

Paul D. Hanna, Dec 01 2009

Keywords

• nonn

Comments

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

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 21*x^3 + 119*x^4 + 703*x^5 +...
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 +...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A001006, A002426, A168597, A168599.

Programs

Magma

m:=30;
A002426:= func< n | (&+[ Binomial(n, k)*Binomial(k, n-k): k in [0..n]]) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!( Exp( (&+[A002426(j)^2*x^j/j: j in [1..m+2]]) ) )); // G. C. Greubel, Mar 16 2021

Mathematica

A002426[n_]:= GegenbauerC[n, -n, -1/2];
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 *)

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))}

Sage

m=30
def A002426(n): return sum( binomial(n, k)*binomial(k, n-k) for k in (0..n) )
def A168598_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp( sum( A002426(j)^2*x^j/j for j in [1..m+2])) ).list()
A168598_list(m) # G. C. Greubel, Mar 16 2021