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A222527
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O.g.f.: Sum_{n>=0} (n^7)^n * exp(-n^7*x) * x^n / n!.
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5
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1, 1, 8191, 1742343625, 2998587019946701, 24204004899040755811870, 666480349285726891499539272955, 50789872166903636182659702516635946082, 9237419992097529135737293866043969707761346313, 3590622358224471993651445012122431990834934483552661750
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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) = Stirling2(7*n, n).
a(n) = [x^(7*n)] (7*n)! * (exp(x) - 1)^n / n!.
a(n) = [x^(6*n)] 1 / Product_{k=1..n} (1-k*x).
a(n) = 1/n! * [x^n] Sum_{k>=0} (k^7)^k*x^k / (1 + k^7*x)^(k+1).
a(n) ~ n^(6*n) * 7^(7*n) / (sqrt(2*Pi*(1-c)*n) * exp(6*n) * (7-c)^(6*n) * c^n), where c = -LambertW(-7*exp(-7)). - Vaclav Kotesovec, May 11 2014
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EXAMPLE
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O.g.f.: A(x) = 1 + x + 8191*x^2 + 1742343625*x^3 + 2998587019946701*x^4 +...+ Stirling2(7*n, n)*x^n +...
where
A(x) = 1 + 1^7*x*exp(-1^7*x) + 2^14*exp(-2^7*x)*x^2/2! + 3^21*exp(-3^7*x)*x^3/3! + 4^28*exp(-4^7*x)*x^4/4! + 5^35*exp(-5^7*x)*x^5/5! +...
is a power series in x with integer coefficients.
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MATHEMATICA
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PROG
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(PARI) {a(n)=polcoeff(sum(k=0, n, (k^7)^k*exp(-k^7*x +x*O(x^n))*x^k/k!), n)}
(PARI) {a(n)=1/n!*polcoeff(sum(k=0, n, (k^7)^k*x^k/(1+k^7*x +x*O(x^n))^(k+1)), n)}
(PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(6*n))), 6*n)}
(PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(7*n, n)}
for(n=0, 12, 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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