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A059116 The sequence lambda(4,n), where lambda is defined in A055203. Number of ways of placing n identifiable positive intervals with a total of exactly four starting and/or finishing points. 4

%I

%S 0,0,6,114,978,6810,43746,271194,1653378,9998970,60229986,362088474,

%T 2174656578,13054316730,78345032226,470127588954,2820937720578,

%U 16926142884090,101558406986466,609355090964634,3656144492925378

%N The sequence lambda(4,n), where lambda is defined in A055203. Number of ways of placing n identifiable positive intervals with a total of exactly four starting and/or finishing points.

%C For all n, a(n)=1*6^n-4*3^n+6*1^n-4*0^n+1*0^n [with 0^0=1] where powers are taken of triangular numbers and multiplied by binomial coefficients with alternating signs.

%H Vincenzo Librandi, <a href="/A059116/b059116.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (10,-27,18).

%F For n>0, a(n) = 6^n-4*3^n+6.

%F a(n) = 10*a(n-1)-27*a(n-2)+18*a(n-3) for n>3. G.f.: -6*x^2*(9*x+1) / ((x-1)*(3*x-1)*(6*x-1)). - _Colin Barker_, Sep 14 2014

%e a(2)=6 since intervals a-a and b-b can be combined as a-a-b-b, a-b-a-b, a-b-b-a, b-a-b-a, b-a-a-b, or b-a-a-b.

%p A059116:=n->`if`(n<2, 0, 6^n-4*3^n+6): seq(A059116(n), n=0..20); # _Wesley Ivan Hurt_, Sep 14 2014

%o (MAGMA) [1*6^n-4*3^n+6*1^n-4*0^n+1*0^n: n in [0..30]]; // _Vincenzo Librandi_, Sep 23 2011

%o (PARI) concat([0,0], Vec(-6*x^2*(9*x+1)/((x-1)*(3*x-1)*(6*x-1)) + O(x^100))) \\ _Colin Barker_, Sep 14 2014

%Y Cf. A058809, A059117.

%K nonn,easy

%O 0,3

%A _Henry Bottomley_, Jan 05 2001

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Last modified October 19 04:40 EDT 2019. Contains 328211 sequences. (Running on oeis4.)