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A304662
Total number of domino tilings of Ferrers-Young diagrams summed over all partitions of 2n.
11
1, 2, 6, 16, 42, 106, 268, 650, 1580, 3750, 8862, 20598, 47776, 109248, 248966, 562630, 1264780, 2823958, 6282198, 13884820, 30590124, 67051982, 146463790, 318588916, 690882926, 1492592450, 3215372064, 6904561416, 14786529836, 31574656096, 67261524262
OFFSET
0,2
FORMULA
a(n) = Sum_{k=0..A304790(n)} k * A304789(n,k).
a(n) = Sum_{k=0..n} A304718(n,k).
a(n) = A296625(n) for n < 7.
EXAMPLE
a(2) = 6:
._. .___. ._._. .___. ._.___. .___.___.
| | |___| | | | |___| | |___| |___|___|
|_| | | |_|_| |___| |_|
| | |_|
|_|
MAPLE
h:= proc(l, f) option remember; local k; if min(l[])>0 then
`if`(nops(f)=0, 1, h(map(x-> x-1, l[1..f[1]]), subsop(1=[][], f)))
else for k from nops(l) while l[k]>0 by -1 do od;
`if`(nops(f)>0 and f[1]>=k, 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-> b(2*n$2, []):
seq(a(n), n=0..12);
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, 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]]];
a[n_] := b[2n, 2n, {}];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Aug 29 2021, after Alois P. Heinz *)
CROSSREFS
Row sums of A304718.
Bisection (even part) of A304680.
Sequence in context: A102699 A266124 A217194 * A296625 A156664 A025169
KEYWORD
nonn
AUTHOR
Alois P. Heinz, May 16 2018
STATUS
approved