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%I #37 Apr 14 2023 07:14:45
%S 1,2,4,14,106,1732,66634,5745700,1058905642,461715853196,
%T 461918527950694,989913403174541980,5009399946447021173140,
%U 60070720443204091719085184,1548154498059133199618813305334,92346622775540905956057053976278584
%N a(n) = Sum_{k=0..n} C(n,k)^k.
%C Row sums of A219206.
%H Reinhard Zumkeller, <a href="/A167008/b167008.txt">Table of n, a(n) for n = 0..75</a>
%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Interesting asymptotic formulas for binomial sums</a>, Jun 09 2013.
%F 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
%t Flatten[{1,Table[Sum[Binomial[n, k]^k, {k,0,n}], {n,20}]}]
%t (* 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 *)
%t Total/@Table[Binomial[n,k]^k,{n,0,20},{k,0,n}] (* _Harvey P. Dale_, Oct 19 2021 *)
%o (PARI) a(n)=sum(k=0,n,binomial(n,k)^k)
%o (Haskell) a167008 = sum . a219206_row -- _Reinhard Zumkeller_, Feb 27 2015
%o (Magma) [(&+[Binomial(n,j)^j: j in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Aug 26 2022
%o (SageMath) [sum(binomial(n,j)^j for j in (0..n)) for n in (0..20)] # _G. C. Greubel_, Aug 26 2022
%Y Cf. A000169, A014062, A167009, A167010, A184731, A219206, A220359, A362288.
%K nonn,nice
%O 0,2
%A _Paul D. Hanna_, Nov 17 2009
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