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A277919 Triangle read by rows: CL(n,k) is the number of labeled subgraphs with k edges of the n-cycle C_n.

%I A277919 #36 Sep 27 2022 10:36:04
%S A277919 1,1,1,3,2,1,7,6,3,1,15,16,10,4,1,31,40,30,15,5,1,63,96,84,50,21,6,1,
%T A277919 127,224,224,154,77,28,7,1,255,512,576,448,258,112,36,8,1,511,1152,
%U A277919 1440,1248,810,405,156,45,9,1,1023,2560,3520,3360,2420,1362,605,210,55,10,1
%N A277919 Triangle read by rows: CL(n,k) is the number of labeled subgraphs with k edges of the n-cycle C_n.
%H A277919 Andrew Howroyd, <a href="/A277919/b277919.txt">Table of n, a(n) for n = 0..1274</a>
%H A277919 Thomas Selig, <a href="https://arxiv.org/abs/2202.06487">Combinatorial aspects of sandpile models on wheel and fan graphs</a>, arXiv:2202.06487 [math.CO], 2022.
%F A277919 The identity CL(n,k) = 2^(n-2*k) * CL(n,n-k) can be proved combinatorially.
%F A277919 G.f.: (1 - 2*x + 2*x^2)/((1-x)*(1 - y*x - 2*x + y*x^2)). - _Andrew Howroyd_, Sep 27 2019
%e A277919 For row 3 of the triangle below: there are 7 labeled subgraphs of the triangle C_3 with 0 edges, 6 with 1 edge, 3 with 2 edges, and 1 with 3 edges (C_3 itself).
%e A277919 Triangle begins:
%e A277919      1;
%e A277919      1,    1;
%e A277919      3,    2,    1;
%e A277919      7,    6,    3,    1;
%e A277919     15,   16,   10,    4,    1;
%e A277919     31,   40,   30,   15,    5,    1;
%e A277919     63,   96,   84,   50,   21,    6,   1;
%e A277919    127,  224,  224,  154,   77,   28,   7,   1;
%e A277919    255,  512,  576,  448,  258,  112,  36,   8,  1;
%e A277919    511, 1152, 1440, 1248,  810,  405, 156,  45,  9,  1;
%e A277919   1023, 2560, 3520, 3360, 2420, 1362, 605, 210, 55, 10, 1;
%e A277919   ...
%o A277919 (PARI) T(n)={[Vecrev(p) | p<-Vec((1 - 2*x + 2*x^2)/((1-x)*(1 - y*x - 2*x + y*x^2)) + O(x*x^n))]}
%o A277919 { my(A=T(12)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Sep 27 2019
%Y A277919 Row sums give A005592.
%Y A277919 Middle diagonal gives A110170.
%K A277919 nonn,tabl
%O A277919 0,4
%A A277919 _John P. McSorley_, Nov 03 2016
%E A277919 More terms from _John P. McSorley_, Nov 17 2016