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A196556 O.g.f.: Sum_{n>=0} 3*(n+3)^(n-1) * x^n / Product_{k=1..n} (1+k*x).

%I A196556 #19 Nov 21 2021 05:05:16
%S A196556 1,3,12,66,483,4476,50454,671649,10328118,180341094,3527385345,
%T A196556 76435691250,1818255212490,47118807865863,1321527658352016,
%U A196556 39889359465259446,1289471521115731611,44450463108654209136,1627806562174453037802
%N A196556 O.g.f.: Sum_{n>=0} 3*(n+3)^(n-1) * x^n / Product_{k=1..n} (1+k*x).
%H A196556 Seiichi Manyama, <a href="/A196556/b196556.txt">Table of n, a(n) for n = 0..399</a>
%F A196556 E.g.f.: exp(-3*LambertW(exp(-x)-1)).
%F A196556 a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*3*(k+3)^(k-1).
%F A196556 E.g.f.: A(x) = G(x)^3 where G(x) = e.g.f. of A058864, which is the number of labeled chordal graphs (connected or not) on n nodes with no induced path P_4.
%F A196556 a(n) ~ 3*sqrt(exp(1)-1) * n^(n-1) / (exp(n-3) * (1-log(exp(1)-1))^(n-1/2)). - _Vaclav Kotesovec_, Jul 09 2013
%e A196556 O.g.f.: A(x) = 1 + 3*x + 12*x^2 + 66*x^3 + 483*x^4 + 4476*x^5 +...
%e A196556 where the o.g.f. is given by:
%e A196556 A(x) = 1 + 3*4^0*x/(1+x) + 3*5^1*x^2/((1+x)*(1+2*x)) + 3*6^2*x^3/((1+x)*(1+2*x)*(1+3*x)) + 3*7^3*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) +...
%e A196556 E.g.f.: A(x) = 1 + 3*x + 12*x^2/2! + 66*x^3/3! + 483*x^4/4! + 4476*x^5/5! +...
%e A196556 where the e.g.f. is given by:
%e A196556 A(x)^(1/3) = 1 + x + 2*x^2/2! + 8*x^3/3! + 49*x^4/4! + 402*x^5/5! + 4144*x^6/6! +...+ A058864(n)*x^n/n! +...
%t A196556 CoefficientList[Series[E^(-3*LambertW[E^(-x)-1]), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jul 09 2013 *)
%o A196556 (PARI) {a(n)=polcoeff(sum(m=0, n, 3*(m+3)^(m-1)*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}
%o A196556 (PARI) /* E.g.f. = G(x)^3 where G(x) = e.g.f. of A058864 */
%o A196556 {A058864(n)=polcoeff(sum(m=0, n, (m+1)^(m-1)*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}
%o A196556 {a(n)=n!*polcoeff(sum(k=0,n,A058864(k)*x^k/k!+x*O(x^n))^3,n)}
%o A196556 (PARI) a(n)=sum(k=0, n, (-1)^(n-k)*stirling(n, k, 2)*3*(k+3)^(k-1));
%o A196556 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-3*lambertw(exp(-x)-1)))) \\ _Seiichi Manyama_, Nov 21 2021
%Y A196556 Cf. A058864, A196555, A196557.
%K A196556 nonn
%O A196556 0,2
%A A196556 _Paul D. Hanna_, Oct 03 2011