A384374 Expansion of e.g.f. exp(x)*(exp(x) - 1)*(exp(x) - x - 1)^2.

0, 0, 0, 0, 0, 30, 390, 3080, 19236, 104874, 524250, 2471172, 11176968, 49065302, 210698670, 890007456, 3712887756, 15342622434, 62938446690, 256735466012, 1042705518960, 4220535078990, 17038468898550, 68644258099320, 276111986410740

Offset: 0

Author

Enrique Navarrete, May 27 2025

Keywords

• nonn
• easy

Comments

a(n) is the number of strings of length n defined on {0, 1, 2, 3} with at least one 1, at least two 2's, at least two 3's, and with no restriction on the number of 0's.

Examples

			a(6) = 390 since the strings are the 180 permutations of 012233, the 90 permutations of 112233, the 60 permutations of 122233, and the 60 permutations of 122333.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (19,-156,726,-2109,3963,-4814,3644,-1560,288).

Crossrefs

Cf. A000478.

Formula

a(n) = 4^n - (2*n + 9)*3^(n-1) + (n^2 + 7*n + 12)*2^(n-2) - (n^2 + n + 1).

G.f.: x^5*(30 - 180*x + 350*x^2 - 224*x^3)/((1 - 4*x)*(1 - 3*x)^2*(1 - 3*x + 2*x^2)^3). - Stefano Spezia, May 27 2025