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A103997 Square array T(M,N) read by antidiagonals: number of dimer tilings of a 2*M X 2*N Moebius strip. 13
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Table of n, a(n) for n=0..54.

W. T. Lu and F. Y. Fu, Dimer statistics on the Moebius strip and the Klein bottle, arXiv:cond-mat/9906154 [cond-mat.stat-mech], 1999.

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

EXAMPLE

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,

  ...

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-Fran├žois Alcover, Dec 03 2018 *)

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, A334178.

Sequence in context: A113711 A339019 A257894 * A256895 A223256 A013561

Adjacent sequences:  A103994 A103995 A103996 * A103998 A103999 A104000

KEYWORD

nonn,tabl

AUTHOR

Ralf Stephan, Feb 26 2005

STATUS

approved

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Last modified March 4 20:38 EST 2021. Contains 341803 sequences. (Running on oeis4.)