OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: A(x) = 3/((1-9*x) + 2*(1-9*x)^(2/3)).
a(n) = 1 + Sum_{m=0..n-1} Sum_{k=0..n-m} C(k,n-m-k)*3^k*(-1)^(n-m-k)*C(n+k,n). - Vladimir Kruchinin, Sep 17 2019
Conjecture: n*(n-1)*a(n) - (19*n-18)*(n-1)*a(n-1) + 9*(11*n^2-31*n+22)*a(n-2) - 9*(3*n-4)*(3*n-5)*a(n-3) = 0. - R. J. Mathar, Nov 16 2012
a(n) ~ 3^(2*n+1)/(2*Gamma(2/3) * n^(1/3))*(1 - sqrt(3)*Gamma(2/3)^2 / (4*Pi*n^(1/3))). - Vaclav Kotesovec, Feb 04 2014
MAPLE
seq(coeff(series(3/((1-9*x) + 2*(1-9*x)^(2/3)), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Sep 17 2019
MATHEMATICA
CoefficientList[Series[3/((1-9*x) + 2*(1-9*x)^(2/3)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 04 2014 *)
PROG
(PARI) a(n)=polcoeff(3/((1-9*x)+2*(1-9*x+x*O(x^n))^(2/3)), n, x)
(Maxima)
a(n):=sum(sum(binomial(k, n-m-k)*3^k*(-1)^(n-m-k)*binomial(n+k, n), k, 0, n-m), m, 0, n-1)+1; /* Vladimir Kruchinin, Sep 09 2019 */
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 3/((1-9*x) + 2*(1-9*x)^(2/3)) )); // G. C. Greubel, Sep 17 2019
(Sage)
def A097189_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( 3/((1-9*x) + 2*(1-9*x)^(2/3)) ).list()
A097189_list(30) # G. C. Greubel, Sep 17 2019
(GAP) List([0..30], n-> 1 + Sum([0..n-1], k-> Sum([0..n-k], j-> (-1)^(n-k-j)*3^j*Binomial(j, n-k-j)*Binomial(n+j, n) )) ); # G. C. Greubel, Sep 17 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 03 2004
STATUS
approved