A336873 - OEIS

%I #29 Sep 01 2022 05:00:15

%S 1,3,73,36729,473940001,155741521320033,1453730786373283012225,

%T 415588116056535702096428038017,

%U 3278068950996636050857475073848209555969,756475486389705843580676191270930552553654909184513,5850304627708628483969594929628923064185219454493588333628772353

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

%H Seiichi Manyama, <a href="/A336873/b336873.txt">Table of n, a(n) for n = 0..37</a>

%H Vaclav Kotesovec, <a href="/A336873/a336873.jpg">Plot of a(n)/(binomial(n + n/sqrt(2),n/sqrt(2)) * binomial(n,n/sqrt(2)))^n for n = 1..500</a>

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

%t a[n_] := Sum[(Binomial[n+k, k] * Binomial[n, k])^n, {k, 0, n} ]; Array[a, 11, 0] (* _Amiram Eldar_, Aug 06 2020 *)

%o (PARI) {a(n) = sum(k=0, n, (binomial(n+k,k)*binomial(n,k))^n)}

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

%o (SageMath)

%o def A336873(n): return sum((binomial(2*j,j)*binomial(n+j, n-j))^n for j in (0..n))

%o [A336873(n) for n in (0..20)] # _G. C. Greubel_, Aug 31 2022

%Y Cf. A001850, A005259, A092813, A092814, A092815, A167010, A218689, A336829, A346200.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Aug 06 2020