A393912 - OEIS

%I #13 Mar 08 2026 10:28:10

%S 1,3,11,42,171,751,3509,17135,86331,445161,2337097,12448148,67095277,

%T 365264609,2005444499,11091616038,61738014131,345581890401,

%U 1944078165249,10985238037766,62322123340849,354849796233957,2027085961125183,11614483110130587,66729581614802773

%N a(n) = 1 + binomial(n+2,3) + Sum_{k=0..n-1} a(k) * a(n-1-k).

%H Vincenzo Librandi, <a href="/A393912/b393912.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f. A(x) satisfies A(x) = 1/(1-x) + x/(1-x)^4 + x*A(x)^2.

%F G.f.: (1 - sqrt(1 - 4*x*(1/(1-x) + x/(1-x)^4)))/(2*x).

%t CoefficientList[Series[(1-Sqrt[1-4*x*(1/(1-x)+x/(1-x)^4)])/(2*x),{x,0,25}],x] (* _Vincenzo Librandi_, Mar 07 2026 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec((1-sqrt(1-4*x*(1/(1-x)+x/(1-x)^4)))/(2*x))

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 25); Coefficients(R! (1 - Sqrt(1 - 4*x*(1/(1-x) + x/(1-x)^4)))/(2*x)); // _Vincenzo Librandi_, Mar 07 2026

%Y Cf. A086616, A393911.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, Mar 02 2026