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A222079
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O.g.f.: Sum_{n>=0} n^n*(n+5)^n * exp(-n*(n+5)*x) * x^n / n!.
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4
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1, 6, 62, 1040, 24076, 703800, 24786512, 1020779520, 48130232528, 2557117300640, 151180506557280, 9846055638729216, 700523098562671360, 54066239308284456960, 4499576117943522662400, 401709919258066014720000, 38299206898825369235170560, 3883999501445283274005895680
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OFFSET
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0,2
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COMMENTS
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Generally, for p>=1, a(n) = 1/n!*Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (k+p)^n = Sum_{j=0..n} binomial(n,j) * p^(n-j) * StirlingS2(n+j,n).
a(n) ~ n^n * 2^(2*n+p/2) / (sqrt(2*Pi*(1-c)*n) * exp(n) * (2-c)^n * c^(n+p/2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599...
(End)
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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+5)^k * x^k / (1 + k*(k+5)*x)^(k+1).
a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (k+5)^n.
a(n) ~ n^n * 2^(2*n+2) / (sqrt(Pi*(1-c)*n) * exp(n) * (2-c)^n * c^(n+5/2)), 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 + 6*x + 62*x^2 + 1040*x^3 + 24076*x^4 + 703800*x^5 +...
where
A(x) = 1 + 6*x*exp(-6*x) + 14^2*exp(-14*x)*x^2/2! + 24^3*exp(-24*x)*x^3/3! + 36^4*exp(-36*x)*x^4/4! + 50^5*exp(-50*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] * 5^(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+5)^m*x^m*exp(-m*(m+5)*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+5)^k*x^k/(1+k*(k+5)*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+5)^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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