%I #17 Feb 19 2021 09:45:40

%S 1,4,10,110,12085,146047246,21329798064184488,

%T 454960285458888331666496499822180,

%U 206988861344833157526906045960863418528301538238377184771619952355

%N a(n+1) = a(n)^2 - (-1)^n * binomial(n+2,2) with a(1) = 1.

%H G. C. Greubel, <a href="/A118378/b118378.txt">Table of n, a(n) for n = 1..12</a>

%H <a href="/index/Aa#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 + ...</a>

%F a(n) = A000290(a(n)) - A033999(n)*A000217(n+1).

%e a(2) = a(1)^2 + (-1)^2 * 2*(2+1)/2 = 1*1 + 1*3 = 4;

%e a(3) = a(2)^2 + (-1)^3 * 3*(3+1)/2 = 4*4 - 3*2 = 10;

%t a[n_]:= a[n]= If[n==1, 1, a[n-1]^2 + (-1)^n*Binomial[n+1, 2]];

%t Table[a[n], {n, 10}] (* _G. C. Greubel_, Feb 18 2021 *)

%o (Sage)

%o @CachedFunction

%o def A118378(n):

%o if (n==1): return 1

%o else: return A118378(n-1)^2 +(-1)^n*binomial(n+1, 2)

%o [A118378(n) for n in (1..10)] # _G. C. Greubel_, Feb 18 2021

%o (Magma)

%o A118378:= func< n | n eq 1 select 1 else Self(n-1)^2 + (-1)^n*Binomial(n+1,2) >;

%o [A118378(n): n in [1..10]];

%o (PARI) a(n) = if (n==1, 1, n--; a(n)^2 - (-1)^n * binomial(n+2,2)); \\ _Michel Marcus_, Feb 19 2021

%Y Cf. A000217, A000290, A033999.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, May 16 2006

%E Corrected by _Don Reble_, Nov 22 2006