A390659 - OEIS

%I #21 Jan 13 2026 08:13:36

%S 1,3,7,21,61,179,527,1561,4617,13667,40455,119765,354541,1049587,

%T 3107183,9198529,27231329,80615763,238655191,706515909,2091572621,

%U 6191900707,18330529343,54265778345,160648643833,475584938515,1407923702391,4168023401829,12339034453917,36528530817411

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

%H Vincenzo Librandi, <a href="/A390659/b390659.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = 1 + 2 * Sum_{k=0..floor((n-1)/2)} a(k) * a(n-1-2*k).

%t a[0]=1; a[n_]:=a[n]=1+2*Sum[a[k]*a[n-1-2*k],{k,0,Floor[(n-1)/2]}];

%t Table[a[n],{n,0,30}] (* _Vincenzo Librandi_, Jan 12 2026 *)

%o (PARI) a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+2*sum(j=0, (i-1)\2, v[j+1]*v[i-2*j])); v;

%o (Magma) a := [1]; for n in [1..30] do s := 0; for k in [0..Floor((n-1)/2)] do s +:= a[k+1] * a[n-2*k]; end for; Append(~a, 1 + 2*s); end for; a; // _Vincenzo Librandi_, Jan 12 2026

%Y Cf. A162326, A390660, A390661.

%Y Cf. A351972.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, Nov 14 2025