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A124812 Number of 4-ary Lyndon words of length n with exactly four 1s. 4

%I

%S 3,21,135,702,3402,15282,65610,270540,1082565,4221639,16120377,

%T 60450138,223205220,813100356,2927177028,10428053400,36804946455,

%U 128817263385,447470664795,1543773631158,5292938720718,18044108743734,61193066237550

%N Number of 4-ary Lyndon words of length n with exactly four 1s.

%F o.g.f. 3 x^5 (1-5 x + 9 x^2 - 6 x^3)/((1-3 x^2)^2 (1- 3 x)^4) = 1/4*((x/(1-3*x))^4 - x^4/(1-3*x^2)^2) a(n) = 1/4*sum_{d|4,d|n} mu(d) C(n/d-1,(n-4)/d )*3^((n-4)/d) = 1/4*C(n-1,3)*3^(n-4) if n is odd = 1/4*C(n-1,3)*3^(n-4) - 1/4*(n/2-1)*3^((n-4)/2) if n is even

%e a(6) = 21 because 1111ab, 1111ba, 111a1b, 111b1a, 11a11b for ab = 23, 24, 34 (accounting for 15 words) and 1111aa, 111a1a for a=2,3,4 (accounting for 6 words) are all Lyndon of length 6

%t 3*(1 - 5*x + 9*x^2 - 6*x^3)/((1 - 3*x)^4*(1 - 3*x^2)^2) + O[x]^23 // CoefficientList[#, x]& (* _Jean-Fran├žois Alcover_, Sep 19 2017 *)

%Y Cf. A124810, A124811, A124813, A124814, A006918, A124722.

%K nonn

%O 5,1

%A _Mike Zabrocki_, Nov 08 2006

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Last modified October 26 08:00 EDT 2021. Contains 348267 sequences. (Running on oeis4.)