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

1, 1, 5, 119, 32707, 69038213, 1309743837515, 206848589180297555, 281897548265847120670891, 3287603007740009094151486257065, 330891681467139744269091005122077348971

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 + 119*x^3 + 32707*x^4 +...
log(A(x)) = x + 9*x^2/2 + 343*x^3/3 + 130321*x^4/4 +...+ A002426(n)^n*x^n/n +...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..45

Crossrefs

Cf. A001006, A002426, A168598, A225328.

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)^j*x^j/j: j in [1..m+2]]) ) )); // G. C. Greubel, Mar 16 2021

Maple

m:=30;
A002426:= n-> add( binomial(n, k)*binomial(k, n-k), k=0..n );
S := series( exp(add(A002426(j)^j*x^j/j, j = 1..m+2)), x, m+1);
seq(coeff(S, x, j), j = 0..m); # G. C. Greubel, Mar 16 2021

Mathematica

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

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))}

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)^j*x^j/j for j in [1..m+2])) ).list()
A168598_list(m) # G. C. Greubel, Mar 16 2021