A272471 Triangle T(n,m) by rows: The number of tatami tilings of a 2 X n grid with dimers and 2*m monomers.

1, 1, 2, 4, 3, 9, 1, 4, 18, 7, 6, 35, 26, 1, 9, 64, 73, 10, 13, 112, 179, 52, 1, 19, 192, 403, 194, 13, 28, 323, 850, 597, 87, 1, 41, 534, 1707, 1624, 408, 16, 60, 872, 3303, 4046, 1532, 131, 1, 88, 1410, 6203, 9428, 4951, 742, 19, 129, 2260, 11366, 20847, 14361, 3308

Offset

1,3

Links

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

A. Erickson, F. Ruskey, J. Woodcock, M. Schurch, Auspicious tatami mat arrangements, arXiv:1103.3309 (2011).

Formula

G.f. x*(y^2 +2*x*y^2 +x^2*y^2 +1 -x^3)/(x^4 -x^3*y^2 -x^3 -x^2*y^2 +x^2 -2*x +1). - R. J. Mathar, May 01 2016

Example

The triangle starts in row n=1 as:
1,1;
2,4;
3,9,1;
4,18,7;
6,35,26,1;
9,64,73,10;
13,112,179,52,1;
19,192,403,194,13;
28,323,850,597,87,1;
41,534,1707,1624,408,16;
60,872,3303,4046,1532,131,1;
88,1410,6203,9428,4951,742,19;
129,2260,11366,20847,14361,3308,184;
189,3596,20407,44194,38369,12472,1223,22;
277,5687,36018,90492,96071,41559,6330,246,1;
406,8946,62648,179982,228224,125942,27382,1878,25;
595,14007,107602,349244,519071,353929,103504,11084,317,1;
872,21842,182800,663470,1138094,935298,352234,54226,2734,28;
1278,33937,307581,1237436,2418496,2348345,1101887,229886,18137,397,1;
1873,52560,513165,2271066,5002459,5646544,3216889,870490,99142,3818,31;
2745,81168,849726,4109303,10106351,13082702,8864264,3008357,466676,28137,486,1;
4023,125022,1397565,7342252,19999147,29352862,23256181,9640660,1949989,170104,5157,34;
5896,192117,2284716,12971722,38856275,64030031,58492346,28994720,7393154,881310,41813,584,1;
8641,294588,3714618,22686720,74268850,136248856,141811450,82607092,25852870,4033500,277165,6778,37;

Mathematica

Select[Flatten[CoefficientList[CoefficientList[Series[x*(y^2 + 2*x*y^2 + x^2*y^2 + 1 - x^3)/(x^4 - x^3*y^2 - x^3 - x^2*y^2 + x^2 - 2*x + 1), {x, 0, 10}, {y, 0, 10}], x], y]], # != 0 &] (* G. C. Greubel, Apr 28 2017 *)

Crossrefs

Cf. A180965 (row sums), A000930 (column m=0), A046741 (without tatami condition).

Sequence in context: A243051 A253791 A021045 * A367443 A155749 A198931

Adjacent sequences: A272468 A272469 A272470 * A272472 A272473 A272474

Keyword

nonn,tabf

Author

R. J. Mathar, Apr 30 2016

Status

approved