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A304710
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Number of partitions of 2n whose Ferrers-Young diagram cannot be tiled with dominoes.
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4
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0, 0, 0, 1, 2, 6, 12, 25, 46, 85, 146, 250, 410, 666, 1053, 1648, 2527, 3840, 5747, 8525, 12496, 18172, 26165, 37408, 53038, 74714, 104502, 145315, 200808, 276030, 377339, 513342, 694925, 936590, 1256670, 1679310, 2234994, 2963430, 3914701, 5153434, 6760937
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OFFSET
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0,5
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REFERENCES
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Also the number of partitions of 2n where the number of odd parts in even positions differs from the number of odd parts in odd positions.
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..10000
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, Young tableau, Diagrams
Index entries for sequences related to dominoes
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FORMULA
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a(n) = A058696(n) - A000712(n) = A000041(2*n) - A000712(n).
a(n) = A144064(2*n,1) - A144064(n,2).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (8*sqrt(3)*n) * (1 - 2/(3^(1/4)*n^(1/4)) - (sqrt(3)/(2*Pi) + Pi/(48*sqrt(3))) / sqrt(n) + (Pi/(6*3^(3/4)) + 15*3^(1/4)/(8*Pi)) / n^(3/4)). - Vaclav Kotesovec, May 25 2018
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EXAMPLE
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a(3) = 1: the Ferrers-Young diagram of 321 cannot be tiled with dominoes because the numbers of white and black squares (when colored like a chessboard) are different but each domino covers exactly one white and one black square:
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a(4) = 2: 32111, 521.
a(5) = 6: 3211111, 32221, 4321, 52111, 541, 721.
a(6) = 12: 321111111, 3222111, 33321, 432111, 5211111, 52221, 54111, 543, 6321, 72111, 741, 921.
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MAPLE
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b:= proc(n, i, p, c) option remember; `if`(n=0, `if`(c=0, 0, 1),
`if`(i<1, 0, b(n, i-1, p, c)+b(n-i, min(n-i, i), -p, c+
`if`(i::odd, p, 0))))
end:
a:= n-> b(2*n$2, 1, 0):
seq(a(n), n=0..50);
# second Maple program:
a:= n-> (p-> p(2*n)-add(p(j)*p(n-j), j=0..n))(combinat[numbpart]):
seq(a(n), n=0..50);
# third Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, add(
numtheory[sigma](j)*b(n-j, k), j=1..n)*k/n)
end:
a:= n-> b(2*n, 1)-b(n, 2):
seq(a(n), n=0..50);
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CROSSREFS
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Column k=0 of A304789.
Cf. A000041, A000712, A058696, A144064, A182616, A304662.
Sequence in context: A294565 A210068 A210633 * A137829 A262196 A261667
Adjacent sequences: A304707 A304708 A304709 * A304711 A304712 A304713
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KEYWORD
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nonn
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AUTHOR
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Alois P. Heinz, May 17 2018
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STATUS
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approved
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