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A162940 a(n) = binomial(n+1,2)*6^2.

%I A162940 #26 Nov 29 2024 21:13:02
%S A162940 0,36,108,216,360,540,756,1008,1296,1620,1980,2376,2808,3276,3780,
%T A162940 4320,4896,5508,6156,6840,7560,8316,9108,9936,10800,11700,12636,13608,
%U A162940 14616,15660,16740,17856,19008,20196,21420,22680,23976,25308,26676,28080,29520,30996
%N A162940 a(n) = binomial(n+1,2)*6^2.
%C A162940 Number of n permutations (n>=2) of 7 objects s, t, u, v, z, x, y with repetition allowed, containing n-2 u's. Example: If n=2 then n-2 = zero (0) u, a(1)=36 because we have ss, st, sv, sz, sx, sy, ts, tt, tv, tz, tx, ty, vs, vt, vv, vz, vx, vy, zs, zt, zv, zz, zx, zy, xs, xt, xv, xz, xx, xy, ys, yt, yv, yz, yx, yy. If n=3 then n-2 = one (1) u, a(2) = 108, >> ssu, stu, etc. If n=4 then n-2 = two (2) u, a(2)= 216, >> ssuu, stuu, ..., txuu, etc. If n=5 then n-2 = three (3) u, a(3)=360, >> ssuuu, stuuu, ..., txuuu, etc.
%H A162940 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A162940 From _Amiram Eldar_, Sep 01 2022: (Start)
%F A162940 Sum_{n>=1} 1/a(n) = 1/18.
%F A162940 Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/9 - 1/18. (End)
%F A162940 From _Amiram Eldar_, Feb 22 2023: (Start)
%F A162940 a(n) = 18*n*(n+1) = 36*A000217(n) = 18*A002378(n).
%F A162940 Product_{n>=1} (1 - 1/a(n)) = -(18/Pi)*cos(sqrt(11)*Pi/6).
%F A162940 Product_{n>=1} (1 + 1/a(n)) = (18/Pi)*cos(sqrt(7)*Pi/6). (End)
%F A162940 From _Elmo R. Oliveira_, Nov 29 2024: (Start)
%F A162940 G.f.: 36*x/(1-x)^3.
%F A162940 E.g.f.: 18*x*(2 + x)*exp(x).
%F A162940 a(n) = 3*A049598(n) = 2*A163758(n).
%F A162940 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%t A162940 Table[Binomial[n + 1, 2]*6^2, {n, 0, 58}]
%o A162940 (PARI) a(n)=18*n*(n+1) \\ _Charles R Greathouse IV_, Jun 16 2017
%Y A162940 Cf. A000217, A002378, A046092, A027468, A035008, A123296.
%Y A162940 Cf. A049598, A163758.
%K A162940 nonn,easy
%O A162940 0,2
%A A162940 _Zerinvary Lajos_, Jul 18 2009, Jul 19 2009