OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
a(n) = Sum_{k = 0..n} C(k,n-k)*3^(k)*(-1)^(n-k)*C(n+k-1,n-1), n>0, a(0)=1.
G.f.: A(x) = 1 + x*B'(x)/B(x), where B(x) = (1-(1-9*x)^(1/3))/(3*x) is the g.f. of A097188.
n*(n-1)*a(n) = 18*(n-1)^2*a(n-1) - 9*(3*n-5)*(3*n-4)*a(n-2). - Vaclav Kotesovec, May 22 2013
a(n) ~ 3^(2*n-1)/(GAMMA(2/3)*n^(1/3)). - Vaclav Kotesovec, May 22 2013
a(n) = (Gamma(n+2/3)/Gamma(2/3)+Gamma(n+1/3)/(Gamma(1/3)))*3^(2*n-1)/ Gamma(n+1)) for n > 0. - Peter Luschny, Jul 05 2013
From Peter Bala, Mar 11 2022: (Start)
a(n) = [x^n] (1/(1 - 3*x + 3*x^2))^n. Cf. A122868(n) = [x^n] (1 + 3*x + 3*x^2)^n.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. (End)
MAPLE
A225439 := n -> `if`(n=0, 1, (GAMMA(n+2/3)/GAMMA(2/3)+GAMMA(n+1/3)/(GAMMA(1/3)))* 3^(2*n-1)/GAMMA(n+1)): seq(A225439(i), i=0..20); # Peter Luschny, Jul 05 2013
MATHEMATICA
Table[Sum[Binomial[k, n-k]*3^k*(-1)^(n-k)*Binomial[n+k-1, n-1], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 22 2013 *)
PROG
(Maxima) a(n):=if n=0 then 1 else sum(binomial(k, n-k)*3^(k)*(-1)^(n-k)*binomial(n+k-1, n-1), k, 0, n);
(PARI) x='x+O('x^66); Vec(3*x/((1-(1-9*x)^(1/3))*(1-9*x)^(2/3))) \\ Joerg Arndt, May 08 2013
(PARI) {a(n)=local(B=(1-(1-9*x+x^2*O(x^n))^(1/3))/(3*x)); polcoeff(1+x*B'/B, n, x)} \\ Paul D. Hanna, May 08 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, May 08 2013
STATUS
approved