A336829 a(n) = Sum_{k=0..n} binomial(n+k,k)^n.

1, 3, 46, 9065, 25561876, 1048567813062, 632156164654144530, 5652307059542612442465921, 755658094192422806457805924637704, 1521188219372604726826961340683399629967888, 46388428590466766659538640978460161019178279424832676

Offset: 0

Author

Ilya Gutkovskiy, Aug 05 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..41

Crossrefs

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

Programs

Magma

[(&+[Binomial(2*n-j,n)^n: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 31 2022

Mathematica

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

PARI

a(n) = sum(k=0, n, binomial(n+k, k)^n); \\ Michel Marcus, Aug 05 2020

SageMath

def A336829(n): return sum(binomial(2*n-j, n)^n for j in (0..n))
[A336829(n) for n in (0..20)] # G. C. Greubel, Aug 31 2022

Formula

a(n) ~ exp(-1/8) * 2^(2*n^2) / (Pi*n)^(n/2). - Vaclav Kotesovec, Jul 10 2021