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 A304710 Number of partitions of 2n whose Ferrers-Young diagram cannot be tiled with dominoes. 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 REFERENCES 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. LINKS 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 FORMULA 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 EXAMPLE 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:    ._____.    |_|X|_|    |X|_|    |_| . 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. MAPLE 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); CROSSREFS 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 KEYWORD nonn AUTHOR Alois P. Heinz, May 17 2018 STATUS approved

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Last modified January 22 04:27 EST 2020. Contains 331133 sequences. (Running on oeis4.)