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A118932
E.g.f.: A(x) = exp( Sum_{n>=0} x^(3^n)/3^((3^n -1)/2) ).
5
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
0,4
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.
LINKS
FORMULA
a(n) = Sum_{k=0..floor(n/3)} (n!/(k!*(n-3*k)!*3^k)) * a(k), with a(0)=1.
EXAMPLE
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! + ...
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 *)
PROG
(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
(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
CROSSREFS
Cf. A118931.
Variants: A118930, A118935.
Sequence in context: A338534 A318843 A001470 * A053499 A218003 A146909
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 06 2006
STATUS
approved