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 A304719 Number of domino tilings of Ferrers-Young diagrams of partitions of 2n using exactly floor(n/2) horizontally oriented dominoes. 1
 1, 1, 2, 5, 14, 28, 62, 150, 380, 787, 1760, 3951, 9338, 19536, 43224, 94326, 213278, 448193, 979712, 2094981, 4622262, 9670378, 20886560, 44067191, 95469402, 198712506 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Table of n, a(n) for n=0..25. Eric Weisstein's World of Mathematics, Ferrers Diagram Wikipedia, Domino Wikipedia, Domino tiling Wikipedia, Ferrers diagram Wikipedia, Mutilated chessboard problem Wikipedia, Partition (number theory) Wikipedia, Polyomino Wikipedia, Young tableau, Diagrams Index entries for sequences related to dominoes FORMULA a(n) = A304718(n,floor(n/2)). EXAMPLE a(3) = 5: : .___. ._.___. .___. ._._. ._._.___. : |___| | |___| |___| | | | | | |___| : | | |_| | | | |_|_| |_|_| : |_| | | |_|_| |___| : | | |_| : |_| MAPLE h:= proc(l, f) option remember; local k; if min(l[])>0 then `if`(nops(f)=0, 1, h(map(u-> u-1, l[1..f[1]]), subsop(1=[][], f))) else for k from nops(l) while l[k]>0 by -1 do od; expand( `if`(nops(f)>0 and f[1]>=k, x*h(subsop(k=2, l), f), 0)+ `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f), 0)) fi end: g:= l-> `if`(add(`if`(l[i]::odd, (-1)^i, 0), i=1..nops(l))=0, `if`(l=[], 1, h([0\$l[1]], subsop(1=[][], l))), 0): b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1\$n]), b(n, i-1, l) +b(n-i, min(n-i, i), [l[], i])): a:= n-> coeff(b(2*n\$2, []), x, iquo(n, 2)): seq(a(n), n=0..14); MATHEMATICA h[l_, f_] := h[l, f] = Module[{k}, If[Min[l] > 0, If[Length[f] == 0, 1, h[l[[1 ;; f[[1]]]] - 1, ReplacePart[f, 1 -> Nothing]]], For[k = Length[l], l[[k]]>0, k--]; If[Length[f] > 0 && f[[1]] >= k, x*h[ReplacePart[l, k -> 2], f], 0] + If[k > 1 && l[[k - 1]] == 0, h[ReplacePart[l, {k -> 1, k - 1 -> 1}], f], 0]]]; g[l_] := If[Sum[If[OddQ[l[[i]]], (-1)^i, 0], {i, 1, Length[l]}] == 0, If[l == {}, 1, h[Table[0, {l[[1]]}], ReplacePart[l, 1 -> Nothing]]], 0]; b[n_, i_, l_] := If[n == 0 || i == 1, g[Join[l, Table[1, {n}]]], b[n, i-1, l] + b[n-i, Min[n-i, i], Append[l, i]]]; T[n_] := CoefficientList[b[2 n, 2 n, {}], x]; a[n_] := T[n][[Floor[n/2] + 1]]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Dec 28 2022, after Alois P. Heinz in A304718 *) CROSSREFS Cf. A304718. Sequence in context: A194124 A349094 A212340 * A022630 A047133 A031874 Adjacent sequences: A304716 A304717 A304718 * A304720 A304721 A304722 KEYWORD nonn AUTHOR Alois P. Heinz, May 17 2018 STATUS approved

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Last modified November 30 23:17 EST 2023. Contains 367463 sequences. (Running on oeis4.)