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A167008
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a(n) = Sum_{k=0..n} C(n,k)^k.
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14
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1, 2, 4, 14, 106, 1732, 66634, 5745700, 1058905642, 461715853196, 461918527950694, 989913403174541980, 5009399946447021173140, 60070720443204091719085184, 1548154498059133199618813305334, 92346622775540905956057053976278584
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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Limit_{n->oo} a(n)^(1/n^2) = (1-r)^(-r/2) = 1.533628065110458582053143..., where r = A220359 = 0.70350607643066243... is the root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Dec 12 2012
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MATHEMATICA
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Flatten[{1, Table[Sum[Binomial[n, k]^k, {k, 0, n}], {n, 20}]}]
(* Program for numerical value of the limit a(n)^(1/n^2) *) (1-r)^(-r/2)/.FindRoot[(1-r)^(2*r-1)==r^(2*r), {r, 1/2}, WorkingPrecision->100] (* Vaclav Kotesovec, Dec 12 2012 *)
Total/@Table[Binomial[n, k]^k, {n, 0, 20}, {k, 0, n}] (* Harvey P. Dale, Oct 19 2021 *)
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PROG
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(PARI) a(n)=sum(k=0, n, binomial(n, k)^k)
(Magma) [(&+[Binomial(n, j)^j: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 26 2022
(SageMath) [sum(binomial(n, j)^j for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 26 2022
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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