A004003 -id:A004003 - cp's OEIS frontend

A187617 Array T(m,n) read by antidiagonals: number of domino tilings of the 2m X 2n grid (m>=0, n>=0).

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 13, 36, 13, 1, 1, 34, 281, 281, 34, 1, 1, 89, 2245, 6728, 2245, 89, 1, 1, 233, 18061, 167089, 167089, 18061, 233, 1, 1, 610, 145601, 4213133, 12988816, 4213133, 145601, 610, 1

Offset: 0

N. J. A. Sloane, Mar 11 2011

A099390 is the main entry for this problem.
The even-indexed rows and columns of the square array in A187596.
Row (and column) 2 is given by A122367. - Nathaniel Johnston, Mar 22 2011

			The array begins:
  1,  1,     1,       1,          1,            1, ...
  1,  2,     5,      13,         34,           89, ...
  1,  5,    36,     281,       2245,        18061, ...
  1, 13,   281,    6728,     167089,      4213133, ...
  1, 34,  2245,  167089,   12988816,   1031151241, ...
  1, 89, 18061, 4213133, 1031151241, 258584046368, ...

Alois P. Heinz, Antidiagonals n = 0..26, flattened
N. Allegra, Exact solution of the 2d dimer model: Corner free energy, correlation functions and combinatorics, arXiv:1410.4131 [cond-mat.stat-mech], 2014. See Table 1.
Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter and Tianyuan Xu, Sandpiles and Dominos, El. J. Comb., 22 (2015), P1.66. See Theorem 15.
Index entries for sequences related to dominoes

A187618 is the triangle version.
Cf. A187596, A099390, A348566.
Main diagonal is A004003. Second and third rows give A001519, A188899.

ft:=(m,n)->
2^(m*n/2)*mul( mul(
(cos(Pi*i/(n+1))^2+cos(Pi*j/(m+1))^2), j=1..m/2), i=1..n/2);
T:=(m,n)->round(evalf(ft(m,n),300));

T[m_, n_] := Product[2(2 + Cos[(2j Pi)/(2m+1)] + Cos[(2k Pi)/(2n+1)]), {j, 1, m}, {k, 1, n}];
Table[T[m-n, n] // Round, {m, 0, 8}, {n, 0, m}] // Flatten (* Jean-François Alcover, Aug 05 2018 *)

default(realprecision, 120);
{T(n, k) = round(prod(a=1, n, prod(b=1, k, 4*cos(a*Pi/(2*n+1))^2+4*cos(b*Pi/(2*k+1))^2)))} \\ Seiichi Manyama, Jan 09 2021

More terms from Nathaniel Johnston, Mar 22 2011

A242861 Triangle T(n,k) by rows: number of ways k dominoes can be placed on an n X n chessboard, k>=0.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 12, 44, 56, 18, 1, 24, 224, 1044, 2593, 3388, 2150, 552, 36, 1, 40, 686, 6632, 39979, 157000, 407620, 695848, 762180, 510752, 192672, 35104, 2180, 1, 60, 1622, 26172, 281514, 2135356, 11785382, 48145820, 146702793, 333518324, 562203148

Offset: 0

Ralf Stephan, May 24 2014

Also, coefficients of the matching-generating polynomial of the n X n grid graph.

In the n-th row there are floor(n^2/2)+1 values.

			Triangle starts:
  1
  1
  1  4   2
  1 12  44   56    18
  1 24 224 1044  2593   3388   2150    552     36
  1 40 686 6632 39979 157000 407620 695848 762180 510752 192672 35104 2180
  ...

Alois P. Heinz, Rows n = 0..14, flattened
Ralf Stephan, Two dominoes on the 3x3 chessboard, illustration of T(3,2)=44.
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
Wikipedia, Matching polynomial
Index entries for sequences related to dominoes
Index entries for sequences related to matchings

Cf. A046741, A096713.

b:= proc(n, l) option remember; local k;
      if n=0 then 1
    elif min(l[])>0 then b(n-1, map(h->h-1, l))
    else for k while l[k]>0 do od; expand(`if`(n>1,
         x*b(n, subsop(k=2, l)), 0) +`if`(k (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$n])):
seq(T(n), n=0..8); # Alois P. Heinz, Jun 01 2014

b[n_, l_List] := b[n, l] =  Module[{k}, Which[n == 0, 1, Min[l]>0, b[n-1, l-1], True, For[k=1, l[[k]]>0, k++]; Expand[If[n>1, x*b[n, ReplacePart[l, k -> 2]], 0] + If[k 1, k + 1 -> 1}]], 0] + b[n, ReplacePart[l, k -> 1]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, Array[0&, n]]]; Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jun 16 2015, after Alois P. Heinz *)

def T(n,k):
   G = Graph(graphs.Grid2dGraph(n,n))
   G.relabel()
   mu = G.matching_polynomial()
   return abs(mu[n^2-2*k])

T(n,1) = A046092(n-1), T(n,2) = A242856(n).
T(n,floor(n^2/2)) = A137308(n), T(2n,2n^2) = A004003(n).
sum(k>=0, T(n,k)) = A210662(n,n) = A028420(n).
T(n,3) = A243206(n), T(n,4) = A243215(n), T(n,5) = A243217(n), T(n,floor(n^2/4)) = A243221(n). - Alois P. Heinz, Jun 01 2014

A304662 Total number of domino tilings of Ferrers-Young diagrams summed over all partitions of 2n.

Original entry on oeis.org

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

Alois P. Heinz, May 16 2018

nonn

			a(2) = 6:
._.  .___.  ._._.  .___.  ._.___.  .___.___.
| |  |___|  | | |  |___|  | |___|  |___|___|
|_|  | |    |_|_|  |___|  |_|
| |  |_|
|_|

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
Gus Wiseman, All 42 domino tilings of integer partitions of 8.
Gus Wiseman, All 106 domino tilings of integer partitions of 10.
Index entries for sequences related to dominoes

Row sums of A304718.
Bisection (even part) of A304680.
Cf. A004003, A296625, A300060, A300789, A304677, A304710, A304790.

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);

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 *)

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.

A300060 Number of domino tilings of the diagram of the integer partition with Heinz number n.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 0, 2, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 3, 0, 3, 1, 1, 0, 0, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 0, 1, 0, 0, 1, 0, 1, 5, 0, 0, 1, 1, 0, 3, 0, 2, 0, 0, 0, 1, 1, 3, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 4, 1, 0, 0, 1, 0, 5, 1, 0, 2, 3, 0, 2, 1, 1, 1, 5, 0, 0, 1, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Feb 23 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Alois P. Heinz, Table of n, a(n) for n = 1..100000
Wikipedia, Domino tiling
Gus Wiseman, The a(91) = 5 domino tilings of (6,4).

Cf. A000085, A000720, A000712, A000898, A001222, A004003, A056239, A099390, A138178, A153452, A238690, A296150, A296188, A299925, A299926, A300056, A300061, A304662.

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$l[1]], subsop(1=[][], l))), 0):
a:= n-> g(sort(map(i-> numtheory[pi](i[1])$i[2], ifactors(n)[2]), `>`)):
seq(a(n), n=1..120);  # Alois P. Heinz, May 22 2018

h[l_, f_] := h[l, f] = Module[{k}, If[Min[l] > 0, If[Length[f] == 0, 1, h[Map[Function[x, x-1], l[[Range @ f[[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];
a[n_] := g[Reverse @ Sort[ Flatten[ Map[ Function[i, Table[PrimePi[i[[1]]], i[[2]]]], FactorInteger[n]]]]];
Array[a, 120] (* Jean-François Alcover, May 28 2018, after Alois P. Heinz *)

A189006 Array A(m,n) read by antidiagonals: number of domino tilings of the m X n grid with upper left corner removed iff m*n is odd, (m>=0, n>=0).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 4, 5, 1, 1, 1, 1, 8, 11, 11, 8, 1, 1, 1, 1, 13, 15, 36, 15, 13, 1, 1, 1, 1, 21, 41, 95, 95, 41, 21, 1, 1, 1, 1, 34, 56, 281, 192, 281, 56, 34, 1, 1, 1, 1, 55, 153, 781, 1183, 1183, 781, 153, 55, 1, 1, 1, 1, 89, 209, 2245, 2415, 6728, 2415, 2245, 209, 89, 1, 1

Offset: 0

Alois P. Heinz, Apr 15 2011

			A(3,3) = 4, because there are 4 domino tilings of the 3 X 3 grid with upper left corner removed:
  . .___. . .___. . .___. . .___.
  ._|___| ._|___| ._| | | ._|___|
  | |___| | | | | | |_|_| |___| |
  |_|___| |_|_|_| |_|___| |___|_|
Array begins:
  1, 1,  1,  1,   1,    1,    1, ...
  1, 1,  1,  1,   1,    1,    1, ...
  1, 1,  2,  3,   5,    8,   13, ...
  1, 1,  3,  4,  11,   15,   41, ...
  1, 1,  5, 11,  36,   95,  281, ...
  1, 1,  8, 15,  95,  192, 1183, ...
  1, 1, 13, 41, 281, 1183, 6728, ...

Alois P. Heinz, Antidiagonals n = 0..75, flattened
Eric Weisstein's World of Mathematics, Perfect Matching
Wikipedia, FKT algorithm
Wikipedia, Matching (graph theory)
Index entries for sequences related to dominoes

Rows m=0+1, 2-12 give: A000012, A000045(n+1), A002530(n+1), A005178(n+1), A189003, A028468, A189004, A028470, A189005, A028472, A210724, A028474.
Main diagonal gives: A189002.
Cf. A099390, A187596, A187616, A187617, A187618, A004003.

with(LinearAlgebra):
A:= proc(m, n) option remember; local i, j, s, t, M;
      if m=0 or n=0 then 1
    elif m1 or j>1 or s=0 then
               if j

A[1, 1] = 1; A[m_, n_] := A[m, n] = Module[{i, j, s, t, M}, Which[m == 0 || n == 0, 1, m < n, A[n, m], True, s = Mod[n*m, 2];M[i_, j_] /; j < i := -M[j, i]; M[, ] = 0; For[i = 1, i <= n, i++, For[j = 1, j <= m, j++, t = (i-1)*m+j-s; If[i > 1 || j > 1 || s == 0, If[j < m, M[t, t+1] = 1]; If[i < n, M[t, t+m] = 1-2*Mod[j, 2]]]]]; Sqrt[Det[Array[M, {n*m-s, n*m-s}]]]]]; Table[Table[A[m, d-m], {m, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Dec 26 2013, translated from Maple *)

A279887 Number of tilings of a sphinx of order n by elementary sphinxes (i.e., sphinxes of order 1).

Original entry on oeis.org

1, 1, 4, 16, 153, 71838, 5965398, 2614508085, 9822629511079, 28751930151895611, 162231215752303027270, 32813942272624544838651213, 1257159787425487037702548758466

Offset: 1

Greg Huber, Dec 21 2016

Sphinx tilings are, by convention, understood to be improper tilings composed of two elementary shapes, order-1 sphinxes, that are mirror images of one another. In other words, one can prove that the tiling of an order-n sphinx requires both L-sphinxes and R-sphinxes (each composed of six equilateral triangles) for any n>1. The sequence terms are based on an initial search-tree method by G. Huber, confirmed and extended by Walter Trump using backtracking and a bit-vector method.
Least-squares fitting indicates a growth law in the form of an exponential of a quadratic in n (i.e., proportional to g^(area), where g is a constant).
a(9) from analysis of the tilings and associated seam factor of two hemisphinxes of order 9 (Walter Trump, personal communication). - Greg Huber, Mar 10 2017
a(10), a(11) from double hemisphinx method described above.

			For n=2, a(2)=1 and this single tiling of an order-2 L-sphinx with three elementary R-sphinxes and one elementary L-sphinx is shown in the Wikiwand link.

A. Martin, "The Sphinx Task Centre Problem" in C. Pritchard (ed.) The Changing Shape of Geometry, Cambridge Univ. Press, 2003, 371-378.

Greg Huber, Craig Knecht, Walter Trump, and Robert M. Ziff, Riddles of the sphinx tilings, arXiv:2304.14388 [cond-mat.stat-mech], 2023.
Greg Huber, Craig Knecht, Walter Trump, and Robert M. Ziff, Entropy and chirality in sphinx tilings, Phys. Rev. Res., 6 (2024), 013227.
J.-Y. Lee and R. V. Moody, Lattice Substitution Systems and Model Sets, arXiv:math/0002019 [math.MG], 2000.
J.-Y. Lee and R. V. Moody, Lattice Substitution Systems and Model Sets, Discrete Comput. Geom., 25 (2001), 173-201.
Mathematics Task Centre, Task166.
Walter Trump, The Dangler Method
University of Bielefeld Tilings, Sphinx.
Wikipedia, Sphinx tiling.
Wikiwand, Sphinx Tiling.

Cf. A004003.

a(9) from Greg Huber, Mar 10 2017
a(10)-a(11) from Greg Huber, May 10 2017
a(11) corrected by Walter Trump, Feb 25 2022
a(12)-a(13) from Walter Trump, Feb 25 2022

A300056 Number of normal standard domino tableaux whose shape is the integer partition with Heinz number n.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 0, 2, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 3, 1, 0, 0, 3, 0, 3, 2, 1, 0, 0, 0, 0, 1, 0, 3, 1, 0, 4, 2, 0, 0, 1, 0, 0, 1, 0, 1, 6, 0, 0, 3, 1, 0, 4, 0, 5, 0, 0, 0, 1, 1, 8, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 6, 4, 0, 0, 1, 0, 6, 1, 0, 6, 5, 0, 6, 3, 1, 2, 10, 0, 0, 1, 0, 0, 0, 0, 0, 8, 1, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Feb 23 2018

nonn

A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers. A standard domino tableau is a generalized Young tableau in which all rows and columns are weakly increasing and all regions are dominos. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(75) = 6 tableaux:
1 2 4   1 2 3   1 2 2   1 1 4   1 1 4   1 1 3
1 2 4   1 2 3   1 3 3   2 3 4   2 2 4   2 2 3
3 3     4 4     4 4     2 3     3 3     4 4

Cf. A000085, A000712, A000898, A001222, A004003, A056239, A099390, A112798, A122111, A138178, A153452, A238690, A296150, A296188, A297388, A299926, A300060, A300061.

A127605 a(n) = 2^(2nn) * Product_{i=1..n} Product_{j=1..n} (sin(iPi/(2n+1))^2 + sin(jPi/(2n+1))^2).

Original entry on oeis.org

1, 6, 500, 463736, 4614756624, 485005220494432, 533978739649683515200, 6129678550595328659594928000, 731483813983605533022316212534132992, 905665520470954445892575061753881157482726912

Offset: 0

Miklos Kristof, Apr 03 2007

nonn

Cf. A004003, A007341, A340052.

for n from 0 to 12 do a[n]:=2^(2*n*n)*product(product(sin(i*Pi/(2*n+1))^2+ sin(j*Pi/(2*n+1))^2,j=1..n),i=1..n) od: seq(round(evalf(a[n],300)),n=0..12);

Table[(2*n+1) * 2^(n*(2*n-1)) * Product[Product[Sin[i*Pi/(2*n + 1)]^2 + Sin[j*Pi/(2*n + 1)]^2, {i, 1, j-1}], {j, 2, n}]^2, {n, 0, 15}] // Round (* Vaclav Kotesovec, Dec 30 2020 *)

a(n) ~ Gamma(1/4) * exp(G*(2*n+1)^2/Pi) / (2^(3/2) * Pi^(3/4) * sqrt(n)), where G is Catalan's constant A006752. - Vaclav Kotesovec, Dec 30 2020

A360499 Number of ways to tile an n X n square using rectangles with distinct dimensions.

Original entry on oeis.org

1, 1, 21, 269, 4489, 82981, 2995185, 118897973

Offset: 1

Scott R. Shannon, Feb 09 2023

All possible tilings are counted, including those identical by symmetry. Note that distinct dimensions means that, for example, a 1 x 3 rectangle can only be used once, regardless of if it lies horizontally or vertically.

			a(1) = 1 as the only way to tile a 1 x 1 square is with a square with dimensions 1 x 1.
a(2) = 1 as the only way to tile a 2 x 2 square is with a square with dimensions 2 x 2.
a(3) = 21. The possible tilings, excluding those equivalent by symmetry, are:
.
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
  |           |   |   |       |   |       |   |   |           |
  +           +   +---+---+---+   +---+---+   +   +---+---+---+
  |           |   |           |   |       |   |   |           |
  +           +   +           +   +       +   +   +           +
  |           |   |           |   |       |   |   |           |
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
.
The first tiling can occur in 1 way, the second in 8 different ways, the third in 8 different ways and the fourth in 4 different ways, giving 21 ways in total.

Cf. A360498 (oblongs), A182275 (not necessarily distinct dimensions), A004003, A099390, A065072, A233320, A230031.

A300789 Heinz numbers of integer partitions whose Young diagram can be tiled by dominos.

Original entry on oeis.org

1, 3, 4, 7, 9, 10, 12, 13, 16, 19, 21, 22, 25, 27, 28, 29, 34, 36, 37, 39, 40, 43, 46, 48, 49, 52, 53, 55, 57, 61, 62, 63, 64, 70, 71, 75, 76, 79, 81, 82, 84, 85, 87, 88, 89, 90, 91, 94, 100, 101, 107, 108, 111, 112, 113, 115, 116, 117, 118, 121, 129, 130, 131

Offset: 1

Gus Wiseman, Mar 12 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

This sequence is conjectured to be the Heinz numbers of integer partitions in which the odd parts appear as many times in even as in odd positions.

			Sequence of integer partitions whose Young diagram can be tiled by dominos begins: (), (2), (11), (4), (22), (31), (211), (6), (1111), (8), (42), (51), (33), (222), (411).

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Solomon W. Golomb, Tiling with polyominoes, Journal of Combinatorial Theory, 1-2 (1966), 280-296.
Wikipedia, Domino tiling

Cf. A000712, A000898, A001405, A004003, A045931, A097613, A099390, A299926, A300056, A300060, A300787, A300788, A304662.

a:= proc(n) option remember; local k; for k from 1+
      `if`(n=1, 0, a(n-1)) while (l-> add(`if`(l[i]::odd,
       (-1)^i, 0), i=1..nops(l))<>0)(sort(map(i->
       numtheory[pi](i[1])$i[2], ifactors(k)[2]))) do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 22 2018

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Total[(-1)^Flatten[Position[primeMS[#],_?OddQ]]]===0&] (* Conjectured *)

cp's OEIS Frontend

A187617 Array T(m,n) read by antidiagonals: number of domino tilings of the 2m X 2n grid (m>=0, n>=0).

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A242861 Triangle T(n,k) by rows: number of ways k dominoes can be placed on an n X n chessboard, k>=0.

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A304662 Total number of domino tilings of Ferrers-Young diagrams summed over all partitions of 2n.

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A300060 Number of domino tilings of the diagram of the integer partition with Heinz number n.

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A189006 Array A(m,n) read by antidiagonals: number of domino tilings of the m X n grid with upper left corner removed iff m*n is odd, (m>=0, n>=0).

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A279887 Number of tilings of a sphinx of order n by elementary sphinxes (i.e., sphinxes of order 1).

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A300056 Number of normal standard domino tableaux whose shape is the integer partition with Heinz number n.

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A127605 a(n) = 2^(2*n*n) * Product_{i=1..n} Product_{j=1..n} (sin(i*Pi/(2*n+1))^2 + sin(j*Pi/(2*n+1))^2).

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A127605 a(n) = 2^(2nn) * Product_{i=1..n} Product_{j=1..n} (sin(iPi/(2n+1))^2 + sin(jPi/(2n+1))^2).