A389286 - OEIS

%I #29 Nov 15 2025 12:12:56

%S 1,1,3,12,57,285,1500,8195,46023,264063,1541227,9121923,54619314,

%T 330265437,2013832533,12369068985,76455159257,475232409786,

%U 2968677006729,18627255774187,117346391111751,741923837211168,4706241730137593,29942623860897282,191028099206837802

%N G.f. A(x) satisfies A(x) = 1 + x/(1-x^3)^2 * A(x)^3.

%H Vincenzo Librandi, <a href="/A389286/b389286.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-5*k-1,k) * A001764(n-3*k).

%t Table[Sum[Binomial[2*n-5*k-1,k]*Binomial[3*(n-3*k),n-3*k]/(2*(n-3*k)+1),{k,0,Floor[n/3]}],{n,0,25}] (* _Vincenzo Librandi_, Nov 15 2025 *)

%o (PARI) a(n) = sum(k=0, n\3, binomial(2*n-5*k-1, k)*binomial(3*(n-3*k), n-3*k)/(2*(n-3*k)+1));

%o (Magma) [&+[Binomial(2*n-5*k-1, k)*Binomial(3*(n-3*k), n-3*k)/(2*(n-3*k)+1): k in [0..Floor(n/3)]] : n in [0..30] ]; // _Vincenzo Librandi_, Nov 15 2025

%Y Cf. A366176, A389284.

%Y Cf. A001764, A390103.

%K nonn,easy

%O 0,3

%A _Seiichi Manyama_, Oct 26 2025