A225439 - OEIS

%I #30 Nov 02 2024 06:49:56

%S 1,3,21,162,1305,10773,90342,765936,6546177,56293380,486451251,

%T 4220183916,36731240910,320571837810,2804298945840,24580601689752,

%U 215832643307217,1898042178972285,16714070686567620,147360883148636850,1300623629653125855

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

%H Vincenzo Librandi, <a href="/A225439/b225439.txt">Table of n, a(n) for n = 0..200</a>

%F 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.

%F 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.

%F 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

%F a(n) ~ 3^(2*n-1)/(GAMMA(2/3)*n^(1/3)). - _Vaclav Kotesovec_, May 22 2013

%F 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

%F From _Peter Bala_, Mar 11 2022: (Start)

%F a(n) = [x^n] (1/(1 - 3*x + 3*x^2))^n. Cf. A122868(n) = [x^n] (1 + 3*x + 3*x^2)^n.

%F 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)

%p 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

%t 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 *)

%o (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);

%o (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

%o (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

%Y Cf. A025748, A097188, A122868.

%K nonn,changed

%O 0,2

%A _Vladimir Kruchinin_, May 08 2013