OFFSET
0,3
COMMENTS
This is the first column of the array A082170.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..150
Valery A. Liskovets, Exact enumeration of acyclic automata, Proc. 15th Conf. "Formal Power Series and Algebr. Combin. (FPSAC'03)", 2003.
Valery A. Liskovets, Exact enumeration of acyclic deterministic automata, Discrete Appl. Math., 154, No.3 (2006), 537-551.
FORMULA
a(n) = a_3(n) where a_3(0) = 1, a_3(n) = Sum_{i=0..n-1} binomial(n, i)*(-1)^(n-i-1)*(i+1)^(3*n-3*i)*a_3(i), n > 0.
1 = Sum_{n>=0} a(n)*exp(-(1+n)^3*x)*x^n/n!. - Vladeta Jovovic, Jul 18 2005
From Paul D. Hanna, May 03 2015: (Start)
1 = Sum_{n>=0} a(n) * x^n/(1 + (n+1)^3*x)^(n+1).
1 = Sum_{n>=0} a(n) * C(n+m-1,n) * x^n/(1 + (n+1)^3*x)^(n+m) for all m>=1.
log(1+x) = Sum_{n>=1} a(n) * x^n/(1 + (n+1)^3*x)^n/n. (End)
MATHEMATICA
a[n_] := If[n == 0, 1, Sum[-(-1)^(n-k) Binomial[n, k] (k+1)^(3(n-k)) a[k], {k, 0, n-1}]];
Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Aug 29 2019 *)
PROG
(PARI) {a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/(1+(k+1)^3*x+x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, May 03 2015
(PARI) {a(n)=if(n==0, 1, sum(k=0, n-1, -(-1)^(n-k)*binomial(n, k)*(k+1)^(3*(n-k))*a(k)))}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, May 03 2015
(Magma)
function a(n) // a = A082158
if n eq 0 then return 1;
else return (&+[Binomial(n, j)*(-1)^(n-j-1)*(j+1)^(3*n-3*j)*a(j): j in [0..n-1]]);
end if;
end function;
[a(n): n in [0..20]]; // G. C. Greubel, Jan 17 2024
(SageMath)
@CachedFunction
def a(n): # A082158
if n==0: return 1
else: return sum(binomial(n, j)*(-1)^(n-j-1)*(j+1)^(3*n-3*j)*a(j) for j in range(n))
[a(n) for n in range(21)] # G. C. Greubel, Jan 17 2024
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Valery A. Liskovets, Apr 09 2003
EXTENSIONS
More terms from Michel Marcus, Aug 29 2019
STATUS
approved