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A222077
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O.g.f.: Sum_{n>=0} n^n*(n+3)^n * exp(-n*(n+3)*x) * x^n / n!.
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4
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1, 4, 34, 504, 10572, 285408, 9419440, 367571200, 16562241744, 846509123520, 48401180913824, 3061687935718272, 212316590908782336, 16018267935253721088, 1306322033185206970368, 114519518777575592865792, 10740222055670467832259840, 1073051903942317493993088000
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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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a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(k+3)^k * x^k / (1 + k*(k+3)*x)^(k+1).
a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (k+3)^n.
a(n) ~ n^n * 2^(2*n+1) / (sqrt(Pi*(1-c)*n) * exp(n) * (2-c)^n * c^(n+3/2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 22 2014
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EXAMPLE
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O.g.f.: A(x) = 1 + 4*x + 34*x^2 + 504*x^3 + 10572*x^4 + 285408*x^5 +...
where
A(x) = 1 + 4*x*exp(-4*x) + 10^2*exp(-10*x)*x^2/2! + 18^3*exp(-18*x)*x^3/3! + 28^4*exp(-28*x)*x^4/4! + 40^5*exp(-40*x)*x^5/5! +...
is a power series in x with integer coefficients.
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MATHEMATICA
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Flatten[{1, Table[Sum[Binomial[n, j] * 3^(n-j) * StirlingS2[n+j, n], {j, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 22 2014 *)
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PROG
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(PARI) {a(n)=polcoeff(sum(m=0, n, m^m*(m+3)^m*x^m*exp(-m*(m+3)*x+x*O(x^n))/m!), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=(1/n!)*polcoeff(sum(k=0, n, k^k*(k+3)^k*x^k/(1+k*(k+3)*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*k^n*(k+3)^n)}
for(n=0, 20, print1(a(n), ", "))
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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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