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 A213658 Irregular triangle read by rows: T(n,k) is the number of dominating subsets with k vertices of the graph G(n) consisting of an edge ab and n triangles, each having one vertex identified with the vertex b. 1
 1, 5, 4, 1, 1, 5, 14, 14, 6, 1, 1, 7, 21, 43, 47, 27, 8, 1, 1, 9, 36, 84, 142, 158, 108, 44, 10, 1, 1, 11, 55, 165, 330, 494, 542, 410, 205, 65, 12, 1, 1, 13, 78, 286, 715, 1287, 1780, 1908, 1527, 875, 346, 90, 14, 1, 1, 15, 105, 455, 1365, 3003, 5005 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Row n contains 2n + 2 entries. Sum of entries in row n = 3^n + 2^{2n+1} = A213659(n). LINKS Table of n, a(n) for n=1..61. S. Alikhani and E. Deutsch, Graphs with domination roots in the right half-plane, arXiv preprint arXiv:1305.3734 [math.CO], 2013-2014. S. Alikhani and Y. H. Peng, Introduction to domination polynomial of a graph, arXiv:0905.2251 [math.CO], 2009. T. Kotek, J. Preen, F. Simon, P. Tittmann, and M. Trinks, Recurrence relations and splitting formulas for the domination polynomial, arXiv:1206.5926 [math.CO], 2012. FORMULA Generating polynomial of row n is x^(n+1)*(2+x)^n + x*(1+x)^(2*n+1); this is the domination polynomial of the graph G(n). T(n,k) = 2^(2*n+1-k)*binomial(n,k-n-1) + binomial(2*n+1,k-1) (n >= 1; 1 <= k <= 2*n+2). EXAMPLE Row 1 is 1,5,4,1 because the graph G(1) is abcd with edges ab, bc, bd, and cd; there is 1 dominating subset of size 1 ({b}); all binomial(4,2)=6 subsets of size 2 of {a,b,c,d} with the exception of {c,d} are dominating; all binomial(4,3)=4 subsets of size 3 of {a,b,c,d} are dominating; obviously, {a,b,c,d} is dominating. Triangle starts: 1, 5, 4, 1; 1, 5, 14, 14, 6, 1; 1, 7, 21, 43, 47, 27, 8, 1; MAPLE T := proc (n, k) options operator, arrow: 2^(2*n+1-k)*binomial(n, k-n-1)+binomial(2*n+1, k-1) end proc: for n to 8 do seq(T(n, k), k = 1 .. 2*n+2) end do; # yields sequence in triangular form MATHEMATICA row[n_] := CoefficientList[x^(n + 1)*(2 + x)^n + x*(1 + x)^(2*n + 1), x] // Rest; Table[row[n], {n, 1, 7}] // Flatten (* Jean-François Alcover, Dec 02 2017 *) PROG (Magma) /* As triangle */ [[2^(2*n+1-k)*Binomial(n, k-n-1) + Binomial(2*n+1, k-1): k in [1..2*n+2]]: n in [1.. 15]]; // Vincenzo Librandi, Jul 21 2019 CROSSREFS Cf. A213659. Sequence in context: A124602 A320060 A132707 * A046575 A154739 A321044 Adjacent sequences: A213655 A213656 A213657 * A213659 A213660 A213661 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Jun 29 2012 STATUS approved

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Last modified September 11 02:28 EDT 2024. Contains 375813 sequences. (Running on oeis4.)