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A272472 Triangle T(n,m) by rows: The number of tatami tilings of a 3 by n grid with dimers and m monomers. 4

%I #15 Feb 13 2022 14:20:24

%S 0,2,0,1,3,0,9,0,1,0,10,0,12,4,0,27,0,13,0,18,0,56,0,16,6,0,75,0,97,0,

%T 18,0,38,0,198,0,152,0,18,10,0,177,0,433,0,214,0,18,0,72,0,570,0,836,

%U 0,282,0,18,16,0,393,0,1517,0,1442,0,354,0,18,0,136

%N Triangle T(n,m) by rows: The number of tatami tilings of a 3 by n grid with dimers and m monomers.

%H A. Erickson, F. Ruskey, J. Woodcock, M. Schurch, <a href="http://arxiv.org/abs/1103.3309">Auspicious tatami mat arrangements</a>, arXiv:1103.3309 (2011).

%F G.f. x *(x^4*y^3 +7*x*y^2 +3*x +2*y +y^3 +x^6*y +3*x^2*y -x^3*y^2 -6*x^4*y -x^2*y^5 +x^2*y^3 +y^3*x^6 -2*y^4*x^5 -2*x^3 -2*x^5 +y^5*x^4 -x^3*y^4 -x^5*y^2 +x^7) / (x^6 +x^5*y -2*x^4*y^2 -2*x^2 -x*y +1). - _R. J. Mathar_, May 01 2016

%e The triangle starts in row n=1 and column m=0 as:

%e 0,2,0,1;

%e 3,0,9,0,1;

%e 0,10,0,12;

%e 4,0,27,0,13;

%e 0,18,0,56,0,16;

%e 6,0,75,0,97,0,18;

%e 0,38,0,198,0,152,0,18;

%e 10,0,177,0,433,0,214,0,18;

%e 0,72,0,570,0,836,0,282,0,18;

%e 16,0,393,0,1517,0,1442,0,354,0,18;

%e 0,136,0,1478,0,3472,0,2292,0,426,0,18;

%e 26,0,829,0,4571,0,7052,0,3410,0,498,0,18;

%e 0,250,0,3554,0,12070,0,13076,0,4808,0,570,0,18;

%e 42,0,1691,0,12479,0,28158,0,22480,0,6494,0,642,0,18;

%e 0,454,0,8108,0,37222,0,59530,0,36308,0,8468,0,714,0,18;

%e 68,0,3359,0,31729,0,97766,0,115948,0,55672,0,10730,0,786,0,18;

%e 0,814,0,17768,0,105238,0,231622,0,210880,0,81708,0,13280,0,858,0,18;

%e 110,0,6537,0,76483,0,306606,0,503348,0,361878,0,115568,0,16118,0,930,0,18;

%e 0,1446,0,37736,0,278626,0,803060,0,1016880,0,590846,0,158404,0,19244,0,1002,0,18;

%e 178,0,12511,0,176833,0,889916,0,1923278,0,1929730,0,924216,0,211368,0,22658,0,1074,0,18;

%e 0,2548,0,78144,0,700670,0,2549216,0,4268026,0,3469042,0,1392996,0,275612,0,26360,0,1146,0,18;

%e 288,0,23617,0,395387,0,2430464,0,6661414,0,8867630,0,5948792,0,2032802,0,352288,0,30350,0,1218,0,18;

%e 0,4460,0,158492,0,1690478,0,7547920,0,16089358,0,17395888,0,9787628,0,2883858,0,442548,0,34628,0,1290,0,18;

%e 466,0,44067,0,860069,0,6319840,0,21344172,0,36292416,0,32446518,0,15527142,0,3990996,0,547544,0,39194,0,1362,0,18;

%Y Cf. A180970 (row sums), A068922 (column m=0), A271786 (column m=1), A272471 (2 by n grid), A100245 (row reversed without tatami condition).

%K tabf,nonn

%O 1,2

%A _R. J. Mathar_, Apr 30 2016

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