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 A243424 Triangle T(n,k) read by rows of number of ways k domicules can be placed on an n X n square (n >= 0, 0 <= k <= floor(n^2/2)). 9
 1, 1, 1, 6, 3, 1, 20, 110, 180, 58, 1, 42, 657, 4890, 18343, 33792, 27380, 7416, 280, 1, 72, 2172, 36028, 362643, 2307376, 9382388, 24121696, 37965171, 34431880, 16172160, 3219364, 170985, 1, 110, 5375, 154434, 2911226, 38049764, 355340561, 2408715568 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS A domicule is either a domino or it is formed by the union of two neighboring unit squares connected via their corners. In a tiling the connections of two domicules are allowed to cross each other. The n-th row gives the coefficients of the matching-generating polynomial of the n X n king graph. - Eric W. Weisstein, Jun 20 2017 LINKS Alois P. Heinz, Rows n = 0..13, flattened Eric Weisstein's World of Mathematics, King Graph Eric Weisstein's World of Mathematics, Matching-Generating Polynomial EXAMPLE T(2,1) = 6: +---+ +---+ +---+ +---+ +---+ +---+ |o-o| | | |o | | o| |o | | o| | | | | || | | || | \ | | / | | | |o-o| |o | | o| | o| |o | +---+ +---+ +---+ +---+ +---+ +---+ T(2,2) = 3: +---+ +---+ +---+ |o-o| |o o| |o o| | | || || | X | |o-o| |o o| |o o| +---+ +---+ +---+ Triangle T(n,k) begins: 1; 1; 1, 6, 3; 1, 20, 110, 180, 58; 1, 42, 657, 4890, 18343, 33792, 27380, 7416, 280; 1, 72, 2172, 36028, 362643, 2307376, 9382388, 24121696, 37965171, ... ... MAPLE b:= proc(n, l) option remember; local d, f, k; d:= nops(l)/2; f:=false; if n=0 then 1 elif l[1..d]=[f\$d] then b(n-1, [l[d+1..2*d][], true\$d]) else for k to d while not l[k] do od; expand(b(n, subsop(k=f, l))+ `if`(k1 and l[k+d+1], x*b(n, subsop(k=f, k+d+1=f, l)), 0)+ `if`(k>1 and n>1 and l[k+d-1], x*b(n, subsop(k=f, k+d-1=f, l)), 0)+ `if`(n>1 and l[k+d], x*b(n, subsop(k=f, k+d=f, l)), 0)+ `if`(k (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [true\$(n*2)])): seq(T(n), n=0..7); MATHEMATICA b[n_, l_] := b[n, l] = Module[{d, f, k}, d = Length[l]/2; f = False; Which[ n == 0, 1, l[[1 ;; d]] == Table[f, d], b[n-1, Join[l[[d+1 ;; 2d]], Table[ True, d]]], True, For[k = 1, !l[[k]], k++]; Expand[b[n, ReplacePart[l, k -> f]] + If[k1 && l[[k+d+1]], x*b[n, ReplacePart[l, {k -> f, k + d + 1 -> f}]], 0] + If[k>1 && n>1 && l[[k + d - 1]], x*b[n, ReplacePart[ l, {k -> f, k + d - 1 -> f}]], 0] + If[n>1 && l[[k + d]], x*b[n, ReplacePart[l, {k -> f, k+d -> f}]], 0] + If[k f, k+1 -> f}]], 0]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, Table[True, 2n]]]; Table[T[n], {n, 0, 7}] // Flatten (* Jean-François Alcover, Jun 09 2018, after Alois P. Heinz *) CROSSREFS Columns k=0-5 give: A000012, A002943(n-1) for n>0, A243464, A243465, A243466, A243467. Row sums give A220638. T(n,floor(n^2/2)) gives A243510. T(n,floor(n^2/4)) gives A243511. Cf. A242861 (the same for dominoes), A239264. Sequence in context: A240264 A119743 A272643 * A182227 A108451 A122178 Adjacent sequences: A243421 A243422 A243423 * A243425 A243426 A243427 KEYWORD nonn,tabf AUTHOR Alois P. Heinz, Jun 04 2014 STATUS approved

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Last modified December 3 22:01 EST 2023. Contains 367540 sequences. (Running on oeis4.)