A387012 Number of ternary strings of length 2*n that have fewer 0's than the combined number of 1's and 2's.

0, 4, 48, 496, 4864, 46464, 436992, 4068096, 37601280, 345733120, 3166363648, 28910051328, 263320698880, 2393742770176, 21726260035584, 196938517118976, 1783247797223424, 16132649384411136, 145839570932465664, 1317564543167102976, 11896996193604993024, 107375816824319901696

Offset

0,2

Links

Table of n, a(n) for n=0..21.

Formula

a(n) = 9^n - Sum_{k=0..n} 2^(n-k)*binomial(2*n,n-k).

G.f.: (sqrt(1-8*x)*(sqrt(1-8*x)+12*x-1)-8*x*(1-9*x))/((1-9*x)*sqrt(1-8*x)*(sqrt(1-8*x)+12*x-1)).

a(n) = A001019(n) - A128418(n).

D-finite with recurrence n*a(n) +(-29*n+28)*a(n-1) +12*(23*n-41)*a(n-2) +432*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Aug 26 2025

Example

a(2) = 48 since the strings of length 4 are the following (number of permutations in parentheses): 1110 (4), 1120 (12), 1220 (12), 2220 (4), 1111 (1), 1112 (4), 1122 (6), 1222 (4), 2222 (1).

Mathematica

a[n_] := 9^n - Sum[2^(n-k) * Binomial[2*n, n-k], {k, 0, n}]; Array[a, 22, 0] (* Amiram Eldar, Aug 16 2025 *)

Crossrefs

Cf. A001019, A128418, A385252.

Sequence in context: A283049 A159903 A099671 * A196963 A220325 A265419

Adjacent sequences: A387009 A387010 A387011 * A387013 A387014 A387015

Keyword

nonn

Author

Enrique Navarrete, Aug 12 2025

Status

approved