A189005 -id:A189005 - cp's OEIS frontend

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).

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 *)

A003779 Number of spanning trees in P_5 x P_n.

Original entry on oeis.org

1, 209, 30305, 4140081, 557568000, 74795194705, 10021992194369, 1342421467113969, 179796299139278305, 24080189412483072000, 3225041354570508955681, 431926215138756947267505, 57847355494807961811035009, 7747424602888405489208931601

Offset: 1

Frans J. Faase

Also number of domino tilings of the 9 X (2n-1) rectangle with upper left corner removed. - Alois P. Heinz, Apr 14 2011

A linear divisibility sequence of order 16; a(n) divides a(m) whenever n divides m. It is the product of two 4th-order linear divisibility sequences A143699 and A241606. - Peter Bala, Apr 26 2014

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

P. Raff, Table of n, a(n) for n = 1..200
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
P. Raff, Spanning Trees in Grid Graphs, arXiv:0809.2551 [math.CO], 2008.
P. Raff, Analysis of the Number of Spanning Trees of P_5 x P_n. Contains sequence, recurrence, generating function, and more.
P. Raff, Analysis of the Number of Spanning Trees of Grid Graphs.
Index to divisibility sequences
Index entries for sequences related to trees
Index entries for linear recurrences with constant coefficients, signature (209, -11936, 274208, -3112032, 19456019, -70651107, 152325888, -196664896, 152325888, -70651107, 19456019, -3112032, 274208, -11936, 209, -1).

A row of A116469. Bisection of A189005.

Cf. A143699, A159764, A241606.

seq(resultant(simplify(ChebyshevU(4,(x-4)*(1/2))), simplify(ChebyshevU(n-1,(1/2)*x)), x), n = 1 .. 14); # Peter Bala, Apr 27 2014

a[n_] := 256^(n-1)*Product[Sin[(h*Pi)/10]^2 + Sin[(k*Pi)/(2*n)]^2, {h, 1, 4}, {k, 1, n-1}]; Table[a[n]//Round, {n, 1, 14}] (* Jean-François Alcover, Apr 28 2014 *)

Vec(-x*(x^14-1440*x^12+26752*x^11-185889*x^10+574750*x^9-708928*x^8+708928*x^6-574750*x^5+185889*x^4-26752*x^3+1440*x^2-1)/(x^16-209*x^15+11936*x^14-274208*x^13+3112032*x^12-19456019*x^11+70651107*x^10-152325888*x^9+196664896*x^8-152325888*x^7+70651107*x^6-19456019*x^5+3112032*x^4-274208*x^3+11936*x^2-209*x+1)+O(x^99)) \\ Charles R Greathouse IV, Nov 13 2015

a(n) = 209 a(n-1)
- 11936 a(n-2)
+ 274208 a(n-3)
- 3112032 a(n-4)
+ 19456019 a(n-5)
- 70651107 a(n-6)
+ 152325888 a(n-7)
- 196664896 a(n-8)
+ 152325888 a(n-9)
- 70651107 a(n-10)
+ 19456019 a(n-11)
- 3112032 a(n-12)
+ 274208 a(n-13)
- 11936 a(n-14)
+ 209 a(n-15)
- a(n-16)
[Modified by Paul Raff, Oct 30 2009]
G.f.: -x(x^14-1440x^12+26752x^11 -185889x^10+574750x^9-708928x^8 +708928x^6-574750x^5+185889x^4 -26752x^3+1440x^2-1) / (x^16-209x^15 +11936x^14 -274208x^13+3112032x^12-19456019x^11 +70651107x^10 -152325888x^9 +196664896x^8 -152325888x^7+70651107x^6 -19456019x^5 +3112032x^4-274208x^3+11936x^2-209x+1).
From Peter Bala, Apr 26 2014: (Start)
a(n) = Resultant(U(4,(x-4)/2),U(n-1,x/2)), where U(n,x) denotes the Chebyshev polynomial of the second kind. The polynomial U(4,(x-4)/2) = 209 - 232*x + 93*x^2 - 16*x^3 + x^4 (see A159764) has zeros z_1 = (9 + sqrt(5))/2, z_2 = (9 - sqrt(5))/2, z_3 = (7 + sqrt(5))/2 and z_4 = (7 - sqrt(5))/2. Thus a(n) = U(n-1,1/2*z_1)*U(n-1,1/2*z_2)*U(n-1,1/2*z_3)*U(n-1,1/2*z_4).
a(n) = A143699(n)*A241606(n). (End)

Recurrence from Faase's web page added by N. J. A. Sloane, Feb 03 2009

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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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A003779 Number of spanning trees in P_5 x P_n.

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