A387358 - OEIS

%I #26 Oct 19 2025 07:28:17

%S 1,1,7,28,131,621,2986,14568,71667,355159,1770007,8862217,44543786,

%T 224619864,1135853824,5757673383,29247575203,148847761323,

%U 758776529977,3873722497342,19802568285331,101353224099561,519312615801679,2663507435418221,13673395654874730

%N a(n) = Sum_{k=0..n} binomial(n,k) * binomial(3*k,n-k).

%H Vincenzo Librandi, <a href="/A387358/b387358.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = [x^n] (1 + x * (1 + x)^3)^n.

%F The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x / (1 + x * (1 + x)^3) ). See A364742.

%F a(n) ~ sqrt(4 + sqrt(3) + 12*sqrt(3/208 + 5/(26*sqrt(3)))) * (1/2 + sqrt(3) + sqrt(17 + 100/(3*sqrt(3)))/2)^n / (2*sqrt(6*Pi*n)). - _Vaclav Kotesovec_, Oct 19 2025

%t Table[Sum[Binomial[n,k]Binomial[3*k,n-k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Oct 08 2025 *)

%o (PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(3*k, n-k));

%o (Magma) [&+[Binomial(n, k) * Binomial(3*k, n-k) : k in [0..n] ]: n in [0..40]]; // _Vincenzo Librandi_, Oct 08 2025

%Y Cf. A002426, A385975, A389225.

%Y Cf. A364742, A378403.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Sep 25 2025