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%I #10 Mar 07 2021 18:43:42
%S 1,1,1,3,9,21,81,351,1233,10249,75841,388411,3733401,33702813,
%T 215375889,1984583511,19181083041,141963117201,1797976123393,
%U 22534941675379,202605151063081,2992764505338021,43182110678814801,445326641624332623
%N E.g.f.: A(x) = exp( Sum_{n>=0} x^(3^n)/3^((3^n -1)/2) ).
%C 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.
%H G. C. Greubel, <a href="/A118932/b118932.txt">Table of n, a(n) for n = 0..490</a>
%F a(n) = Sum_{k=0..floor(n/3)} (n!/(k!*(n-3*k)!*3^k)) * a(k), with a(0)=1.
%e E.g.f. A(x) = exp( x + x^3/3 + x^9/3^4 + x^27/3^13 + x^81/3^40 + ...)
%e = 1 + 1*x + 1*x^2/2! + 3*x^3/3! + 9*x^4/4! + 21*x^5/5! + 81*x^6/6! + ...
%t a[n_]:= a[n]= If[n==0, 1, Sum[n!*a[k]/(3^k*k!*(n-3*k)!), {k, 0, Floor[n/3]}] ];
%t Table[a[n], {n, 0, 25}] (* _G. C. Greubel_, Mar 07 2021 *)
%o (PARI) {a(n) = if(n==0,1,sum(k=0,n\3,n!/(k!*(n-3*k)!*3^k)*a(k)))}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) /* Defined by E.G.F.: */
%o {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)}
%o for(n=0,30,print1(a(n),", "))
%o (Sage)
%o @CachedFunction
%o def a(n):
%o f=factorial;
%o if n==0: return 1
%o else: return sum( f(n)*a(k)/(3^k*f(k)*f(n-3*k)) for k in (0..n/3))
%o [a(n) for n in (0..25)] # _G. C. Greubel_, Mar 07 2021
%o (Magma)
%o function a(n)
%o F:=Factorial;
%o if n eq 0 then return 1;
%o else return (&+[F(n)*a(j)/(3^j*F(j)*F(n-3*j)): j in [0..Floor(n/3)]]);
%o end if; return a; end function;
%o [a(n): n in [0..25]]; // _G. C. Greubel_, Mar 07 2021
%Y Cf. A118931.
%Y Variants: A118930, A118935.
%K nonn
%O 0,4
%A _Paul D. Hanna_, May 06 2006
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