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A304710 Number of partitions of 2n whose Ferrers-Young diagram cannot be tiled with dominoes.

%I A304710 #34 Feb 16 2025 08:33:54
%S A304710 0,0,0,1,2,6,12,25,46,85,146,250,410,666,1053,1648,2527,3840,5747,
%T A304710 8525,12496,18172,26165,37408,53038,74714,104502,145315,200808,276030,
%U A304710 377339,513342,694925,936590,1256670,1679310,2234994,2963430,3914701,5153434,6760937
%N A304710 Number of partitions of 2n whose Ferrers-Young diagram cannot be tiled with dominoes.
%C A304710 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.
%H A304710 Alois P. Heinz, <a href="/A304710/b304710.txt">Table of n, a(n) for n = 0..10000</a>
%H A304710 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FerrersDiagram.html">Ferrers Diagram</a>
%H A304710 Wikipedia, <a href="https://en.wikipedia.org/wiki/Domino_(mathematics)">Domino</a>
%H A304710 Wikipedia, <a href="https://en.wikipedia.org/wiki/Domino_tiling">Domino tiling</a>
%H A304710 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ferrers_diagram">Ferrers diagram</a>
%H A304710 Wikipedia, <a href="https://en.wikipedia.org/wiki/Mutilated_chessboard_problem">Mutilated chessboard problem</a>
%H A304710 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%H A304710 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau#Diagrams">Young tableau, Diagrams</a>
%H A304710 <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>
%F A304710 a(n) = A058696(n) - A000712(n) = A000041(2*n) - A000712(n).
%F A304710 a(n) = A144064(2*n,1) - A144064(n,2).
%F A304710 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
%e A304710 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:
%e A304710    ._____.
%e A304710    |_|X|_|
%e A304710    |X|_|
%e A304710    |_|
%e A304710 .
%e A304710 a(4) = 2: 32111, 521.
%e A304710 a(5) = 6: 3211111, 32221, 4321, 52111, 541, 721.
%e A304710 a(6) = 12: 321111111, 3222111, 33321, 432111, 5211111, 52221, 54111, 543, 6321, 72111, 741, 921.
%p A304710 b:= proc(n, i, p, c) option remember; `if`(n=0, `if`(c=0, 0, 1),
%p A304710       `if`(i<1, 0, b(n, i-1, p, c)+b(n-i, min(n-i, i), -p, c+
%p A304710       `if`(i::odd, p, 0))))
%p A304710     end:
%p A304710 a:= n-> b(2*n$2, 1, 0):
%p A304710 seq(a(n), n=0..50);
%p A304710 # second Maple program:
%p A304710 a:= n-> (p-> p(2*n)-add(p(j)*p(n-j), j=0..n))(combinat[numbpart]):
%p A304710 seq(a(n), n=0..50);
%p A304710 # third Maple program:
%p A304710 b:= proc(n, k) option remember; `if`(n=0, 1, add(
%p A304710       numtheory[sigma](j)*b(n-j, k), j=1..n)*k/n)
%p A304710     end:
%p A304710 a:= n-> b(2*n, 1)-b(n, 2):
%p A304710 seq(a(n), n=0..50);
%t A304710 b[n_, i_, p_, c_] := b[n, i, p, c] = 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[OddQ[i], p, 0]]]];
%t A304710 a[n_] := b[2n, 2n, 1, 0];
%t A304710 Table[a[n], {n, 0, 50}]
%t A304710 (* second program: *)
%t A304710 a[n_] := PartitionsP[2n] - Sum[PartitionsP[j]* PartitionsP[n - j], {j, 0, n}];
%t A304710 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Feb 13 2023, after _Alois P. Heinz_ *)
%Y A304710 Column k=0 of A304789.
%Y A304710 Cf. A000041, A000712, A058696, A144064, A182616, A304662.
%K A304710 nonn
%O A304710 0,5
%A A304710 _Alois P. Heinz_, May 17 2018