cp's OEIS Frontend

A336188 a(n) = Sum_{k=0..n} n^k * binomial(n,k)^n.

%I A336188 #49 Aug 27 2022 08:14:13
%S A336188 1,2,13,352,38401,16971876,29359436149,207003074670848,
%T A336188 5679112509686022145,636468045901197095750500,
%U A336188 277939985126193076692203962501,494649880078824954885176565423811200,3447375085398645453825889951638344722092289,97424105704407389799712313421357308088296084669504
%N A336188 a(n) = Sum_{k=0..n} n^k * binomial(n,k)^n.
%H A336188 Seiichi Manyama, <a href="/A336188/b336188.txt">Table of n, a(n) for n = 0..57</a>
%H A336188 Vaclav Kotesovec, <a href="/A336188/a336188.jpg">Plot of a(n) / (2^((n+1)*(2*n-1)/2) * n^(log(n)/8) / Pi^((n-1)/2)) for n = 1..10000000</a>
%F A336188 Let f(n) = 2^((n+1)*(2*n-1)/2) * n^(log(n)/8) / Pi^((n-1)/2). For sufficiently large n 0.7675... < a(n)/f(n) < 0.7900... - _Vaclav Kotesovec_, Jul 11 2020
%F A336188 The above bounds of _Vaclav Kotesovec_ can be recast as: |a(n)/f(n) - exp(-1/4)| <= (3*Pi)^(-2) for sufficiently large n. - _Peter Luschny_, Jul 12 2020
%F A336188 a(n) ~ exp(-1/4) * QPochhammer(exp(-4)) * QPochhammer(-n*exp(-2), exp(-4)) * 2^(n^2 + n/2) / Pi^(n/2) if n is even and a(n) ~ exp(-3/4) * QPochhammer(exp(-4)) * QPochhammer(-n*exp(-4), exp(-4)) * 2^(n^2 + n/2) * sqrt(n)  / Pi^(n/2) if n is odd. - _Vaclav Kotesovec_, Jul 13 2020
%t A336188 Unprotect[Power]; 0^0 = 1; a[n_] := Sum[n^k * Binomial[n, k]^n, {k, 0, n} ]; Array[a, 14, 0] (* _Amiram Eldar_, Jul 11 2020 *)
%o A336188 (PARI) {a(n) = sum(k=0, n, n^k*binomial(n, k)^n)}
%o A336188 (Magma) [(&+[n^j*Binomial(n,j)^n: j in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Aug 26 2022
%o A336188 (SageMath) [sum(n^j*binomial(n,j)^n for j in (0..n)) for n in (0..20)] # _G. C. Greubel_, Aug 26 2022
%Y A336188 Main diagonal of A336187.
%Y A336188 Cf. A167010, A187021, A234971, A241247, A336202, A336213, A336214.
%K A336188 nonn,nice
%O A336188 0,2
%A A336188 _Seiichi Manyama_, Jul 11 2020