A103997 Square array T(M,N) read by antidiagonals: number of dimer tilings of a 2*M X 2*N Moebius strip.

1, 1, 1, 1, 3, 1, 1, 11, 7, 1, 1, 41, 71, 18, 1, 1, 153, 769, 539, 47, 1, 1, 571, 8449, 17753, 4271, 123, 1, 1, 2131, 93127, 603126, 434657, 34276, 322, 1, 1, 7953, 1027207, 20721019, 46069729, 10894561, 276119, 843, 1, 1, 29681, 11332097, 714790675, 4974089647, 3625549353, 275770321, 2226851, 2207, 1

Offset: 0

Author

Ralf Stephan, Feb 26 2005

Keywords

• nonn
• tabl

Examples

			Array begins:
  1,   1,     1,        1,          1,             1,               1,
  1,   3,     7,       18,         47,           123,             322,
  1,  11,    71,      539,       4271,         34276,          276119,
  1,  41,   769,    17753,     434657,      10894561,       275770321,
  1, 153,  8449,   603126,   46069729,    3625549353,    289625349454,
  1, 571, 93127, 20721019, 4974089647, 1234496016491, 312007855309063,
  ...
		

Links

• Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter and Tianyuan Xu, Sandpiles and Dominos, El. J. Comb., 22 (2015), P1.66. See Theorem 18.
• W. T. Lu and F. Y. Wu, Dimer statistics on the Moebius strip and the Klein bottle, arXiv:cond-mat/9906154 [cond-mat.stat-mech], 1999.
• Index entries for sequences related to dominoes

Crossrefs

Rows include A005248, A103998.

Columns 1..7 give A001835(n+1), A334135, A334179, A334180, A334181, A334182, A334183.

Main diagonal gives A334124.

Cf. A099390, A103999, A341741, A334178, A256045, A348566.

Programs

Mathematica

T[M_, N_] := Product[4Sin[(4n-1)Pi/(4N)]^2 + 4Cos[m Pi/(2M+1)]^2, {n, 1, N}, {m, 1, M}];
Table[T[M - N, N] // Round, {M, 0, 9}, {N, 0, M}] // Flatten (* Jean-François Alcover, Dec 03 2018 *)

Formula

T(M, N) = Product_{m=1..M} (Product_{n=1..N} 4*sin(Pi*(4*n-1)/(4*N))^2 + 4*cos(Pi*m/(2*M + 1))^2).

For k > 0, T(n,k) = 2^n * sqrt(Resultant(U_{2*n}(x/2), T_{2*k}(i*x/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1). - Seiichi Manyama, Apr 15 2020