cp's OEIS Frontend

A283614 T(n,k) = number of circular arrays of n 1's, n -1's, and k 0's such that no two adjacent elements are equal.

%I A283614 #17 Apr 03 2017 02:57:53
%S A283614 1,2,6,4,2,10,24,28,12,2,14,56,132,180,132,40,2,18,100,352,804,1196,
%T A283614 1120,600,140,2,22,156,728,2324,5196,8160,8840,6300,2660,504,2,26,224,
%U A283614 1300,5320,15844,34872,56848,67900,57820,33264,11592,1848,2,30,304,2108,10512,39064,110480,240288,402556,515844,496944,348600
%N A283614 T(n,k) = number of circular arrays of n 1's, n -1's, and k 0's such that no two adjacent elements are equal.
%C A283614 The array is circular in the sense that the first and last elements are adjacent.
%C A283614 For linear arrays see A283613.
%F A283614 G.f.: 2*(x*y+1)/sqrt((1-y)*(1-(2*x+1)^2*y))-1.
%F A283614 T(n,0)  G.f.: (1+y)/(1-y).
%F A283614 T(n,1)  G.f.: 2*y*(3-y)/(1-y)^2.
%F A283614 T(n,2)  G.f.: 4*y*(1+3*y-y^2)/(1-y)^3.
%F A283614 T(n,3)  G.f.: 4*y^2*(1+y)*(7-2*y)/(1-y)^4.
%F A283614 T(n,4)  G.f.: 4*y^2*(3+30*y+6*y^2-4*y^3)/(1-y)^5.
%F A283614 T(n,5)  G.f.: 4*y^3*(33+101*y-8*y^3)/(1-y)^6.
%F A283614 T(n,n) = A110707(n).
%F A283614 T(n,2*n) = 2*binomial(2*n,n).
%F A283614 Sum_{2*n+k = m} T(n,k) = A265118(m), m > 3.
%e A283614 The table starts with columns k=0..10 and rows n=0..5:
%e A283614   | 0  1   2   3    4    5    6    7    8    9  10
%e A283614 -----------------------------------------------------------------
%e A283614 0 | 1
%e A283614 1 | 2  6   4
%e A283614 2 | 2 10  24  28   12
%e A283614 3 | 2 14  56 132  180  132   40
%e A283614 4 | 2 18 100 352  804 1196 1120  600  140
%e A283614 5 | 2 22 156 728 2324 5196 8160 8840 6300 2660 504
%e A283614 For n=2, k=3, the 28 arrays are:
%e A283614 [+0-0+0-] [+0+0-0-] [0-+0+0-] [0-0+0+-]
%e A283614 [0+-0+0-] [0+0-+0-] [0+0-0+-] [0+0+-0-]
%e A283614 [-0-0+0+] [-0+0-0+] [0-+0-0+] [0-0-+0+]
%e A283614 [0-0+-0+] [0-0+0-+] [0+-0-0+] [0+0-0-+]
%e A283614 [-+0-0+0] [-+0+0-0] [-0-+0+0] [-0+-0+0]
%e A283614 [-0+0-+0] [-0+0+-0] [+-0-0+0] [+-0+0-0]
%e A283614 [+0-+0-0] [+0-0-+0] [+0-0+-0] [+0+-0-0]
%t A283614 nmax=8; Flatten[CoefficientList[Series[CoefficientList[Series[2*(x*y + 1)/Sqrt[(1 - y)*(1 - (2*x + 1)^2*y)] - 1, {y, 0, nmax }], y], {x, 0, 2nmax + 1 }], x]] (* _Indranil Ghosh_, Apr 02 2017 *)
%Y A283614 Cf. A110707, A265118, A283613.
%K A283614 nonn,tabf
%O A283614 0,2
%A A283614 _Stefan Hollos_, Apr 01 2017