OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Wolfdieter Lang, On generalizations of the Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Thomas M. Richardson, The Super Patalan Numbers, arXiv preprint arXiv:1410.5880 [math.CO], 2014 and J. Int. Seq. 18 (2015) # 15.3.3.
FORMULA
G.f.: 3 / (2+(1-9*x)^(1/3)).
a(n) = Sum_{m=1..n-1} (m/n) * Sum_{k=1..n-m} binomial(k,n-m-k) * 3^k * (-1)^(n-m-k) * binomial(n+k-1,n-1) + 1. - Vladimir Kruchinin, Feb 08 2011
Conjecture: n*(n-1)*a(n) -(n-1)*(19*n-36)*a(n-1) +9*(11*n^2-51*n+60)*a(n-2) -9*(3*n-7)*(3*n-8)*a(n-3) = 0. - R. J. Mathar, Nov 14 2011
a(n) ~ 9^n/(4*Gamma(2/3)*n^(4/3)). - Vaclav Kotesovec, Oct 08 2012
a(n) = (-1)^(n+1) * 3^(2*n+1) * Sum_{k>=0} (-1/2)^(k+1) * binomial(k/3,n). - Seiichi Manyama, Aug 04 2024
a(n) = (1/3)*(4+Sum_{k=0..n} (C(2/3,k)-2*C(1/3,k))*(-9)^k). - Tani Akinari, Jan 19 2026
MAPLE
A025756 := proc(n)
coeftayl( 3/(2+(1-9*x)^(1/3)), x=0, n);
end proc:
seq(A025756(n), n=0..30); # Wesley Ivan Hurt, Aug 02 2014
MATHEMATICA
Table[SeriesCoefficient[3/(2+(1-9*x)^(1/3)), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 08 2012 *)
PROG
(Maxima) a[0]:1$ a[n]:=(1/n)*((9*n-6)*a[n-1]-2*sum(a[k]*a[n-1-k], k, 0, n-1))$ makelist(a[n], n, 0, 1000); /* Tani Akinari, Aug 02 2014 */
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!( 3/(2+(1-9*x)^(1/3)) )); // G. C. Greubel, Jan 01 2026
(SageMath)
def A025756_list(prec):
P.<x>= PowerSeriesRing(QQ, prec)
return P( 3/(2+(1-9*x)^(1/3)) ).list()
print(A025756_list(41)) # G. C. Greubel, Jan 01 2026
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved