This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A385252 #15 Aug 12 2025 19:43:48
%S A385252 0,0,32,432,4608,45440,432896,4051712,37535744,345470976,3165315072,
%T A385252 28905857024,263303921664,2393675661312,21725991600128,
%U A385252 196937443377152,1783243502256128,16132632204541952,145839502212988928,1317564268289196032,11896995094093365248,107375812426273390592
%N A385252 Number of ternary strings of length 2*n that have at least one 0 but less 0's than the combined number of 1's and 2's.
%F A385252 a(n) = 9^n - 4^n - Sum_{k=0..n} 2^(n-k)*C(2*n,n-k) for n > 0.
%F A385252 G.f.: (5*x*(sqrt(1-8*x))*(sqrt(1-8*x)+12*x-1)-8*x*(36*x^2-13*x+1))/(sqrt(1-8*x)*(sqrt(1-8*x)+12*x-1)*(36*x^2-13*x+1)) + 1.
%F A385252 a(n) = A001019(n) - A000302(n) - A128418(n), n > 0.
%F A385252 Conjecture D-finite with recurrence n*a(n) +(-37*n+36)*a(n-1) +4*(131*n-245)*a(n-2) +16*(-221*n+605)*a(n-3) +192*(59*n-213)*a(n-4) +6912*(-2*n+9)*a(n-5)=0. - _R. J. Mathar_, Jul 31 2025
%F A385252 a(n) = 9^n - 4^n - 2^n*binomial(2*n, n)*hypergeom([1, -n], [1+n], -1/2) for n > 0. - _Stefano Spezia_, Aug 05 2025
%e A385252 a(2)=32 since the strings of length 4 are (number of permutations in parentheses): 1110 (4), 1120 (12), 1220 (12), 2220 (4).
%e A385252 a(3)=432 since the strings of length 6 are (number of permutations in parentheses): 111110 (6), 111120 (30), 111220 (60), 112220 (60), 122220 (30), 222220 (6), 001111 (15), 001112 (60), 001122 (90), 001222 (60), 002222 (15).
%t A385252 a[0]=0; a[n_]:=9^n - 4^n - Sum[2^(n-k)*Binomial[2n,n-k],{k,0,n}]; Array[a,22,0] (* _Stefano Spezia_, Jul 31 2025 *)
%Y A385252 Cf. A000302, A001019, A128418, A386670.
%K A385252 nonn
%O A385252 0,3
%A A385252 _Enrique Navarrete_, Jul 28 2025