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A196557
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O.g.f.: Sum_{n>=0} 4*(n+4)^(n-1) * x^n / Product_{k=1..n} (1+k*x).
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4
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1, 4, 20, 128, 1036, 10308, 122560, 1701092, 27053556, 485683128, 9723771156, 214934627476, 5201286731560, 136818097071820, 3888121468512308, 118737900886653664, 3878569457507036988, 134960059001226137588, 4984357865462772982112
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OFFSET
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0,2
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LINKS
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FORMULA
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E.g.f.: exp(-4*LambertW(exp(-x)-1)).
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*4*(k+4)^(k-1).
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.
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
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EXAMPLE
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O.g.f.: A(x) = 1 + 4*x + 20*x^2 + 128*x^3 + 1036*x^4 + 10308*x^5 +...
where the o.g.f. is given by:
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.g.f.: A(x) = 1 + 4*x + 20*x^2/2! + 128*x^3/3! + 1036*x^4/4! + 10308*x^5/5! +...
where the e.g.f. is given by:
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! +...
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! +...
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MATHEMATICA
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CoefficientList[Series[E^(-4*LambertW[E^(-x)-1]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)
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PROG
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(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)}
(PARI) /* E.g.f. = G(x)^4 where G(x) = e.g.f. of A058864 */
{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)}
{a(n)=n!*polcoeff(sum(k=0, n, A058864(k)*x^k/k!+x*O(x^n))^4, n)}
(PARI) a(n)=sum(k=0, n, (-1)^(n-k)*stirling(n, k, 2)*4*(k+4)^(k-1));
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-4*lambertw(exp(-x)-1)))) \\ Seiichi Manyama, Nov 21 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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