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A229259
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O.g.f.: Sum_{n>=0} n! * n^n * x^n / Product_{k=1..n} (1 - n^2*k*x).
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8
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1, 1, 9, 259, 15789, 1693771, 287145789, 71487432619, 24798142070109, 11518873418467051, 6945333793188487869, 5301472723402989073579, 5018547949600497090304029, 5790959348524892656227425131, 8026963462960378548022418765949, 13197920271743736945902641688868139
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} k^(2*n-k) * k! * Stirling2(n, k).
E.g.f.: Sum_{n>=0} (exp(n^2*x) - 1)^n / n^n.
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EXAMPLE
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O.g.f.: A(x) = 1 + x + 9*x^2 + 259*x^3 + 15789*x^4 + 1693771*x^5 +...
where
A(x) = 1 + x/(1-x) + 2!*2^2*x^2/((1-2^2*1*x)*(1-2^2*2*x)) + 3!*3^3*x^3/((1-3^2*1*x)*(1-3^2*2*x)*(1-3^2*3*x)) + 4!*4^4*x^4/((1-4^2*1*x)*(1-4^2*2*x)*(1-4^2*3*x)*(1-4^2*4*x)) +...
Exponential Generating Function.
E.g.f.: E(x) = 1 + x + 9*x^2/2! + 259*x^3/3! + 15789*x^4/4! + 1693771*x^5/5! +...
where
E(x) = 1 + (exp(x)-1) + (exp(4*x)-1)^2/2^2 + (exp(9*x)-1)^3/3^3 + (exp(16*x)-1)^4/4^4 + (exp(25*x)-1)^5/5^5 +...
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MATHEMATICA
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Flatten[{1, Table[Sum[k^(2*n-k) * k! * StirlingS2[n, k], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 08 2014 *)
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PROG
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(PARI) {a(n)=polcoeff(sum(m=0, n, m!*m^m*x^m/prod(k=1, m, 1-m^2*k*x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=n!*polcoeff(sum(m=0, n, (exp(m^2*x+x*O(x^n))-1)^m/m^m), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=sum(k=0, n, k^(2*n-k) * k! * stirling(n, k, 2))}
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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