|
%I #17 Nov 21 2021 05:05:26
%S 1,4,20,128,1036,10308,122560,1701092,27053556,485683128,9723771156,
%T 214934627476,5201286731560,136818097071820,3888121468512308,
%U 118737900886653664,3878569457507036988,134960059001226137588,4984357865462772982112
%N O.g.f.: Sum_{n>=0} 4*(n+4)^(n-1) * x^n / Product_{k=1..n} (1+k*x).
%H Seiichi Manyama, <a href="/A196557/b196557.txt">Table of n, a(n) for n = 0..399</a>
%F E.g.f.: exp(-4*LambertW(exp(-x)-1)).
%F a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*4*(k+4)^(k-1).
%F E.g.f.: A(x) = G(x)^4 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 a(n) ~ 4*sqrt(exp(1)-1)*n^(n-1)/(exp(n-4)*(1-log(exp(1)-1))^(n-1/2)). - _Vaclav Kotesovec_, Jul 09 2013
%e O.g.f.: A(x) = 1 + 4*x + 20*x^2 + 128*x^3 + 1036*x^4 + 10308*x^5 +...
%e where the o.g.f. is given by:
%e A(x) = 1 + 4*5^0*x/(1+x) + 4*6^1*x^2/((1+x)*(1+2*x)) + 4*7^2*x^3/((1+x)*(1+2*x)*(1+3*x)) + 4*8^3*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) +...
%e E.g.f.: A(x) = 1 + 4*x + 20*x^2/2! + 128*x^3/3! + 1036*x^4/4! + 10308*x^5/5! +...
%e where the e.g.f. is given by:
%e A(x)^(1/2) = 1 + 2*x + 6*x^2/2! + 28*x^3/3! + 186*x^4/4! + 1614*x^5/5! + 17332*x^6/6! +...+ A196555(n)*x^n/n! +...
%e A(x)^(1/4) = 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 CoefficientList[Series[E^(-4*LambertW[E^(-x)-1]), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jul 09 2013 *)
%o (PARI) {a(n)=polcoeff(sum(m=0, n, 4*(m+4)^(m-1)*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}
%o (PARI) /* E.g.f. = G(x)^4 where G(x) = e.g.f. of A058864 */
%o {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 {a(n)=n!*polcoeff(sum(k=0,n,A058864(k)*x^k/k!+x*O(x^n))^4,n)}
%o (PARI) a(n)=sum(k=0, n, (-1)^(n-k)*stirling(n, k, 2)*4*(k+4)^(k-1));
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-4*lambertw(exp(-x)-1)))) \\ _Seiichi Manyama_, Nov 21 2021
%Y Cf. A058864, A196555, A196556.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Oct 03 2011
| 