A225439 Expansion of 3*x/((1-(1-9*x)^(1/3))*(1-9*x)^(2/3)).

1, 3, 21, 162, 1305, 10773, 90342, 765936, 6546177, 56293380, 486451251, 4220183916, 36731240910, 320571837810, 2804298945840, 24580601689752, 215832643307217, 1898042178972285, 16714070686567620, 147360883148636850, 1300623629653125855

Offset: 0

Author

Vladimir Kruchinin, May 08 2013

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A025748, A097188, A122868.

Programs

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 *)

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

my(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

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)