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.

0, 0, 32, 432, 4608, 45440, 432896, 4051712, 37535744, 345470976, 3165315072, 28905857024, 263303921664, 2393675661312, 21725991600128, 196937443377152, 1783243502256128, 16132632204541952, 145839502212988928, 1317564268289196032, 11896995094093365248, 107375812426273390592

Offset: 0

Author

Enrique Navarrete, Jul 28 2025

Keywords

• nonn

Examples

			a(2)=32 since the strings of length 4 are (number of permutations in parentheses): 1110 (4), 1120 (12), 1220 (12), 2220 (4).
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).
		

Crossrefs

Cf. A000302, A001019, A128418, A386670.

Programs

Mathematica

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 *)

Formula

a(n) = 9^n - 4^n - Sum_{k=0..n} 2^(n-k)*C(2*n,n-k) for n > 0.

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.

a(n) = A001019(n) - A000302(n) - A128418(n), n > 0.

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

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