A182107 -id:A182107 - cp's OEIS frontend

A226300 a(n) = A182107(4n).

2, 10, 114, 1322, 16428, 214660, 2897424, 40046134, 563527294, 8042361426, 116083167058, 1691193906828, 24830916046462, 366990100477712, 5454733737618820, 81475220265330158, 1222209564554562110, 18404140554678144630, 278069131820486935046, 4214088001120913287256, 64037903464421065585244

Offset: 1

N. J. A. Sloane, Jun 06 2013

nonn

Alejandro Erickson, Frank Ruskey, Enumerating maximal tatami mat coverings of square grids with v vertical dominoes, arXiv:1304.0070 [math.CO], 2013.

Cf. A182107, A226301.

S[0, 0] = 1; S[0, ] = 0; S[n, k_] /; k < 0 || k > Binomial[n + 1, 2] = 0; S[n_, k_] := S[n, k] = S[n - 1, k] + S[n - 1, k - n];
b[n_] := 2 Sum[Sum[k2 = (n^2 - n)/4 - (n - i - 1) - k1; S[n - i - 2, k1]* S[i - 1, k2], {k1, 0, (n^2 - n)/4 - (n - i - 1)}] + Sum[k2 = (n^2 - n)/4; S[Floor[(n - 2)/2], k1]*S[Floor[(n - 2)/2], k2], {k1, 0, (n^2 - n)/4}], {i, 1, Floor[(n - 1)/2]}];
a[n_] := b[4n];
Table[a[n], {n, 1, 21}] (* Jean-François Alcover, Feb 23 2019 *)

A226301 a(n) = A182107(4n+1).

Original entry on oeis.org

2, 20, 210, 2460, 31122, 410378, 5575682, 77445152, 1093987598, 15660579168, 226608224226, 3308255447206, 48658330768786, 720224064591558, 10718841444208526, 160283814975116386, 2406806389622598056, 36273856567768931782, 548495166665709003794, 8318227159058988730096

Offset: 1

N. J. A. Sloane, Jun 06 2013

nonn

Alejandro Erickson, Frank Ruskey, Enumerating maximal tatami mat coverings of square grids with v vertical dominoes, arXiv:1304.0070 [math.CO], 2013.

Cf. A182107, A226300.

S[0, 0] = 1; S[0, ] = 0; S[n, k_] /; k < 0 || k > Binomial[n + 1, 2] = 0; S[n_, k_] := S[n, k] = S[n - 1, k] + S[n - 1, k - n];
b[n_] := 2 Sum[Sum[k2 = (n^2 - n)/4 - (n - i - 1) - k1; S[n - i - 2, k1] * S[i - 1, k2], {k1, 0, (n^2 - n)/4 - (n - i - 1)}] + Sum[k2 = (n^2 - n)/4; S[Floor[(n - 2)/2], k1] * S[Floor[(n - 2)/2], k2], {k1, 0, (n^2 - n)/4}], {i, 1, Floor[(n - 1)/2]}];
a[n_] := b[4n+1];
Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Feb 23 2019 *)

A182110 Irregular triangle read by rows: generating function counting rotationally distinct n X n tatami tilings with n monomers and exactly k vertical dimers.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 2, 1, 2, 3, 6, 4, 2, 2, 1, 2, 3, 6, 9, 8, 7, 6, 2, 2, 2, 1, 2, 3, 6, 9, 14, 15, 14, 14, 10, 8, 6, 4, 2, 2, 2, 1, 2, 3, 6, 9, 14, 22, 24, 25, 28, 25, 22, 19, 14, 10, 10, 8, 4, 4, 2, 2, 2, 1, 2, 3, 6, 9, 14, 22, 32, 37, 42, 49, 48, 49, 46, 38, 34, 30, 24, 20, 16, 12, 12, 10, 6, 4, 4, 2, 2, 2

Offset: 0

Alejandro Erickson, Apr 12 2012

Monomer-dimer tatami tilings are arrangements of 1 X 1 monomers, 2 X 1 vertical dimers and 1 X 2 horizontal dimers on subsets of the integer grid, with the property that no four tiles meet at any point. a(n) applies to tilings of this type which have monomers in their top corners.

a(n) is the table T(2,0); T(3,0), T(3,1); T(4,0), T(4,1), T(4,2), T(4,3); T(5,0), T(5,1) ... where T(n,k) is the number of n X n tilings of the type described above with exactly k vertical dimers when n is even and exactly k horizontal dimers when n is odd.

			T_5(z) = 1 + 2*z + 3*z^2 + 6*z^3 + 4*z^4 + 2*z^5 + 2*z^6;
T(5,2) = 3, and the tilings are as follows:
._ _ _ _ _.
|_|_ _| |_|
|_ _| |_| |
|_| |_| |_|
| |_| |_| |
|_|_|_|_|_|
.
._ _ _ _ _.
|_| |_ _|_|
| |_| |_ _|
|_| |_| |_|
| |_| |_| |
|_|_|_|_|_|
.
._ _ _ _ _.
|_| |_| |_|
| |_| |_| |
|_| |_| |_|
|_|_| |_|_|
|_ _|_|_ _|
The triangle begins:
1
1,2
1,2,3,2
1,2,3,6,4,2,2
1,2,3,6,9,8,7,6,2,2,2
1,2,3,6,9,14,15,14,14,10,8,6,4,2,2,2
1,2,3,6,9,14,22,24,25,28,25,22,19,14,10,10,8,4,4,2,2,2
1,2,3,6,9,14,22,32,37,42,49,48,49,46,38,34,30,24,20,16,12,12,10,6,4,4,2,2,2
1,2,3,6,9,14,22,32,46,56,66,78,84,90,92,88,81,76,69,58,51,44,38,34,28,22,20,16,14,12,8,6,4,4,2,2,2
...

Alejandro Erickson, Table of n, a(n) for n = 0..9999
Alejandro Erickson, Table of coefficients of T_n(z)
Alejandro Erickson, Frank Ruskey, Enumerating maximal tatami mat coverings of square grids with v vertical dominoes, arXiv:1304.0070 [math.CO], 2013.

S_k(z) is entry A053632.
T_n(z) is a partition of A001787(n)/4.
Tatami tilings with the same number of vertical and horizontal dimers is A182107.

@cached_function
def S(n,z):
    out = 1
    for i in [j+1 for j in range(n)]:
        out = out*(1+z^i)
    return out
T = lambda n,z: 2*sum([S(n-i-2,z)*S(i-1,z)*z^(n-i-1) for i in range(1,floor((n-1)/2)+1)]) + S(floor((n-2)/2),z)^2
ZP. = PolynomialRing(ZZ)
#call T(n,x) for the g.f. T_n(x)

G.f.: T_n(z) = Sum_{k>=0} T(n,k)*z^k is equal to

T_n(z) = 2*Sum_{i=1..floor((n-1)/2)} S_{n-i-2}(z)*S_{i-1}(z)*z^{n-i-1} + (S_{floor((n-2)/2))^2, where S_k(z) = Product_{i=1..k} (1+z^i). Note that deg(T_n(z)) = binomial(n-1,2).

Entry revised by N. J. A. Sloane, Jun 06 2013

cp's OEIS Frontend

A226300 a(n) = A182107(4n).

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A226301 a(n) = A182107(4n+1).

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A182110 Irregular triangle read by rows: generating function counting rotationally distinct n X n tatami tilings with n monomers and exactly k vertical dimers.

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