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A336204 a(n) = Sum_{k=0..n} 2^k * binomial(n,k)^n.

%I A336204 #23 Sep 01 2022 05:00:36
%S A336204 1,3,13,171,7761,1256283,741398869,1609036666443,13118066779885825,
%T A336204 399221556627301207443,46476897754761801245056293,
%U A336204 20377119057713827002258336842283,34592895120825704155462768381947657489,222457046333769635263635086646525921070978443
%N A336204 a(n) = Sum_{k=0..n} 2^k * binomial(n,k)^n.
%H A336204 Seiichi Manyama, <a href="/A336204/b336204.txt">Table of n, a(n) for n = 0..59</a>
%F A336204 a(n) ~ c * 2^(n*(n+1)) / (Pi*n)^(n/2), where c = exp(-1/4) * Sum_{k = -oo..oo} 2^k * exp(-2*k^2) = 1.0434092897163574491113380912895917... if n is even and c = exp(-1/4) * Sum_{k = -oo..oo} 2^(k + 1/2) * exp(-2*(k + 1/2)^2) = 1.029587234777114329090639723058125257... if n is odd. - _Vaclav Kotesovec_, Jul 12 2020
%t A336204 Table[Sum[2^k*Binomial[n, k]^n, {k, 0, n}], {n, 0, 15}] (* _Vaclav Kotesovec_, Jul 12 2020 *)
%o A336204 (PARI) {a(n) = sum(k=0, n, 2^k*binomial(n, k)^n)};
%o A336204 (Magma) [(&+[2^j*Binomial(n,j)^n: j in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Aug 31 2022
%o A336204 (SageMath)
%o A336204 def A336204(n): return sum(2^k*binomial(n,k)^n for k in (0..n))
%o A336204 [A336204(n) for n in (0..20)] # _G. C. Greubel_, Aug 31 2022
%Y A336204 Main diagonal of A336203.
%Y A336204 Cf. A167010, A336188, A336202, A336212.
%K A336204 nonn
%O A336204 0,2
%A A336204 _Seiichi Manyama_, Jul 11 2020