A118932 E.g.f.: A(x) = exp( Sum_{n>=0} x^(3^n)/3^((3^n -1)/2) ).

1, 1, 1, 3, 9, 21, 81, 351, 1233, 10249, 75841, 388411, 3733401, 33702813, 215375889, 1984583511, 19181083041, 141963117201, 1797976123393, 22534941675379, 202605151063081, 2992764505338021, 43182110678814801, 445326641624332623

Offset: 0

Author

Paul D. Hanna, May 06 2006

Keywords

• nonn

Comments

Equals invariant column vector V that satisfies matrix product A118931*V = V, where A118931(n,k) = n!/(k!*(n-3*k)!*3^k) for n>=3*k>=0; thus a(n) = Sum_{k=0..floor(n/3)} A118931(n,k)*a(k), with a(0) = 1.

Examples

			E.g.f. A(x) = exp( x + x^3/3 + x^9/3^4 + x^27/3^13 + x^81/3^40 + ...)
= 1 + 1*x + 1*x^2/2! + 3*x^3/3! + 9*x^4/4! + 21*x^5/5! + 81*x^6/6! + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..490

Crossrefs

Cf. A118931.

Variants: A118930, A118935.

Programs

Magma

function a(n)
     F:=Factorial;
      if n eq 0 then return 1;
    else return (&+[F(n)*a(j)/(3^j*F(j)*F(n-3*j)): j in [0..Floor(n/3)]]);
      end if; return a; end function;
[a(n): n in [0..25]]; // G. C. Greubel, Mar 07 2021

Mathematica

a[n_]:= a[n]= If[n==0, 1, Sum[n!*a[k]/(3^k*k!*(n-3*k)!), {k, 0, Floor[n/3]}] ];
Table[a[n], {n, 0, 25}] (* G. C. Greubel, Mar 07 2021 *)

PARI

{a(n) = if(n==0,1,sum(k=0,n\3,n!/(k!*(n-3*k)!*3^k)*a(k)))}
for(n=0,30,print1(a(n),", "))

PARI

/* Defined by E.G.F.: */
{a(n) = n!*polcoeff( exp(sum(k=0,ceil(log(n+1)/log(3)),x^(3^k)/3^((3^k-1)/2))+x*O(x^n)),n,x)}
for(n=0,30,print1(a(n),", "))

Sage

@CachedFunction
def a(n):
    f=factorial;
    if n==0: return 1
    else: return sum( f(n)*a(k)/(3^k*f(k)*f(n-3*k)) for k in (0..n/3))
[a(n) for n in (0..25)] # G. C. Greubel, Mar 07 2021

Formula

a(n) = Sum_{k=0..floor(n/3)} (n!/(k!*(n-3*k)!*3^k)) * a(k), with a(0)=1.