OFFSET
1,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..200
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
FORMULA
E.g.f.: exp(1-(1-3*x)^(1/3)) - 1, if one takes a(0)=0.
a(n) = 6*(n-2)*a(n-1) - (3*n-8)*(3*n-7)*a(n-2) + a(n-3), a(0)=1, a(1)=1, a(2)=3.
a(n) = 1 + (n-1)!*Sum_{m=1..n} ( Sum_{k=1..n-m} C(k, n-m-k)*C(k+n-1, n-1)*(-1/3)^(n-m-k) ) / (m-1)!, n > 1. - Vladimir Kruchinin, Aug 08 2010
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator 1/(1-x)^2*d/dx. Cf. A001515, A016036 and A028575. - Peter Bala, Nov 25 2011
E.g.f. with offset 0: exp(1-(1-3*x)^(1/3))/(1-3*x)^(2/3). - Sergei N. Gladkovskii, Jul 07 2012.
a(n) ~ sqrt(2*Pi)*3^(n-1)*exp(1-n)*n^(n-5/6)/Gamma(2/3) * (1-sqrt(3)*Gamma(2/3)^2/(2*Pi*n^(1/3))). - Vaclav Kotesovec, Aug 10 2013
MATHEMATICA
a[1]=1; a[n_]:= 1 +(n-1)!*Sum[Binomial[k, n-m-k]*Binomial[k+n-1, n-1]*(-1/3)^(n-m-k)/(m-1)!, {m, n}, {k, n-m}]; Table[a[n], {n, 20}] (* Jean-François Alcover, Jul 05 2013, after Vladimir Kruchinin *)
Rest@With[{m=30}, CoefficientList[Series[Exp[1-Surd[1-3*x, 3]] -1, {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, Oct 02 2023 *)
PROG
(Maxima) a(n):=if n=1 then 1 else (n-1)!*sum(sum(binomial(k, n-m-k)* (-1/3)^(n-m-k)*binomial(k+n-1, n-1), k, 1, n-m)/(m-1)!, m, 1, n)+1; /* Vladimir Kruchinin, Aug 08 2010 */
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(1-(1-3*x)^(1/3)) - 1 ))); // G. C. Greubel, Oct 02 2023
(SageMath)
def A015735_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( exp(1-(1-3*x)^(1/3)) -1 ).egf_to_ogf().list()
a=A015735_list(40); a[1:] # G. C. Greubel, Oct 02 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved