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A196555 O.g.f.: Sum_{n>=0} 2*(n+2)^(n-1) * x^n / Product_{k=1..n} (1+k*x). 5

%I

%S 1,2,6,28,186,1614,17332,222254,3317326,56532264,1083571422,

%T 23081180918,541047188936,13843339479298,383952455939662,

%U 11475711580482268,367729128426998450,12577206203908139494,457341567152354085700,17619050162270848917366

%N O.g.f.: Sum_{n>=0} 2*(n+2)^(n-1) * x^n / Product_{k=1..n} (1+k*x).

%H Seiichi Manyama, <a href="/A196555/b196555.txt">Table of n, a(n) for n = 0..399</a>

%F E.g.f.: exp(-2*LambertW(exp(-x)-1)).

%F a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*2*(k+2)^(k-1).

%F a(n) = Sum_{k=0..n} C(n,k)*A058864(n-k)*A058864(k); exponential convolution 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) ~ 2*sqrt(exp(1)-1) * n^(n-1) / (exp(n-2) * (1-log(exp(1)-1))^(n-1/2)). - _Vaclav Kotesovec_, Jul 09 2013

%e O.g.f.: A(x) = 1 + 2*x + 6*x^2 + 28*x^3 + 186*x^4 + 1614*x^5 +...

%e where the o.g.f. is given by:

%e A(x) = 1 + 2*3^0*x/(1+x) + 2*4^1*x^2/((1+x)*(1+2*x)) + 2*5^2*x^3/((1+x)*(1+2*x)*(1+3*x)) + 2*6^3*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) +...

%e E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 28*x^3/3! + 186*x^4/4! + 1614*x^5/5! +...

%e where the e.g.f. is given by:

%e A(x)^(1/2) = 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^(-2*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, 2*(m+2)^(m-1)*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}

%o (PARI) {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) = sum(k=0,n,binomial(n,k)*A058864(n-k)*A058864(k))}

%o (PARI) a(n)=sum(k=0, n, (-1)^(n-k)*stirling(n, k, 2)*2*(k+2)^(k-1));

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-2*lambertw(exp(-x)-1)))) \\ _Seiichi Manyama_, Nov 21 2021

%Y Cf. A058864, A196556, A196557.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 03 2011

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Last modified August 7 23:50 EDT 2022. Contains 355995 sequences. (Running on oeis4.)