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

%I #17 Sep 01 2022 05:00:25

%S 1,3,46,9065,25561876,1048567813062,632156164654144530,

%T 5652307059542612442465921,755658094192422806457805924637704,

%U 1521188219372604726826961340683399629967888,46388428590466766659538640978460161019178279424832676

%N a(n) = Sum_{k=0..n} binomial(n+k,k)^n.

%H Seiichi Manyama, <a href="/A336829/b336829.txt">Table of n, a(n) for n = 0..41</a>

%F a(n) ~ exp(-1/8) * 2^(2*n^2) / (Pi*n)^(n/2). - _Vaclav Kotesovec_, Jul 10 2021

%t Table[Sum[Binomial[n + k, k]^n, {k, 0, n}], {n, 0, 10}]

%o (PARI) a(n) = sum(k=0, n, binomial(n+k, k)^n); \\ _Michel Marcus_, Aug 05 2020

%o (Magma) [(&+[Binomial(2*n-j,n)^n: j in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Aug 31 2022

%o (SageMath)

%o def A336829(n): return sum(binomial(2*n-j, n)^n for j in (0..n))

%o [A336829(n) for n in (0..20)] # _G. C. Greubel_, Aug 31 2022

%Y Cf. A001700, A112028, A112029, A167010, A219562, A219563, A219564.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Aug 05 2020