A389060 - OEIS

%I #14 Sep 25 2025 16:40:09

%S 1,0,2,9,18,65,245,735,2506,8784,29277,100320,347589,1193049,4123821,

%T 14318759,49682490,172830228,602551604,2102511006,7346864653,

%U 25706596155,90032376546,315635979373,1107596435917,3889779452940,13671032884875,48082756125105

%N a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(3*k,n-2*k).

%H Vincenzo Librandi, <a href="/A389060/b389060.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = [x^n] (1 + x^2 * (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^2 * (1 + x)^3) ). See A389063.

%t Table[Sum[Binomial[n,k]*Binomial[3*k,n-2*k],{k,0,Floor[n/2]}],{n,0,30}] (* _Vincenzo Librandi_, Sep 25 2025 *)

%o (PARI) a(n) = sum(k=0, n\2, binomial(n, k)*binomial(3*k, n-2*k));

%o (Magma) [&+[Binomial(n, k)*Binomial(3*k, n-2*k): k in [0..Floor(n/2)]]: n in [0..30]]; // _Vincenzo Librandi_, Sep 25 2025

%Y Cf. A378406, A389063.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Sep 22 2025