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

0,3

Comments

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

Links

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

Example

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 +...

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 *)

Prog

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

Crossrefs

Cf. A001006, A002426, A168597, A168599.

Sequence in context: A041321 A082428 A015558 * A002711 A218962 A124311

Adjacent sequences: A168595 A168596 A168597 * A168599 A168600 A168601

Keyword

nonn

Author

Paul D. Hanna, Dec 01 2009

Status

approved