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%I #15 Jan 02 2023 12:30:52
%S 0,3,0,4,0,1,6,0,35,0,26,0,1,0,18,0,56,0,16,3,0,52,0,64,0,7,0,10,0,88,
%T 0,80,2,0,60,0,182,0,81,0,8,0,160,0,320,0,96,2,0,102,0,500,0,449,0,
%U 112,0,18,0,340,0,952,0,600,0,120,4,0,184,0,1056
%N Triangle T(n,m) by rows: the number of tatami tilings of the 5 X n floor with dimers and m monomers.
%H R. J. Mathar, <a href="http://list.seqfan.eu/oldermail/seqfan/2016-March/016246.html">Re: Tatami</a>, Seqfan mailing list of Mar 24 2016.
%F G.f. x*( -3*x^11*y^8 +7*x^4*y^7 +x^2*y^7 -24*x^6*y^7 -2*x^5*y^4 -y^5 -2*x^3*y^2 -16*x^9 -9*x^ 2*y +3*x^3 +13*x^5 +7*x^15 -6*x -3*y -22*x*y^4 -32*x*y^2 +16*x^11*y^6 -6*x^13*y^2 -14*x^ 15*y^4 +3*x^13 +14*x^5*y^6 -11*x^11 +6*x^7 -55*y^3*x^10 -17*x^2*y^3 +37*x^6*y -24*y^3*x^14 -48*x^10*y +2*x^17*y^2 -19*x^12*y -4*y^3 -55*x^12*y^3 -8*x^11*y^4 +51*x^8*y^5 +28*x^7*y^6 +72*x^6*y^3 +31*y^2*x^15 -58*x^13*y^4 +x^17 -64*x^11*y^2 +36*x^10*y^5 +55*x^8*y^3 -50*x^6*y^5 -4*x^4*y^3 -5*x^9*y^4 -5*x^8*y +60*y^2*x^5 +84*x^7*y^2 -12*x^9*y^6 +55*x^7*y^4 -2*x^12*y^5 -79*x^9*y^2 -2*x^16*y^3 +4*x^7*y^8 +11*x^4*y -20*x^4*y^5 +6*x^13*y^ 6 +15*x^16*y +17*x^14*y^5 -5*x^10*y^7 -26*x^8*y^7 +x^8*y^9 +21*y*x^14 +11*x^2*y^5 +x^7* y^10 -x^10*y^9 +14*x^3*y^4 +9*x^3*y^6 -5*y^7*x^12 +3*x^9*y^8) / (x^14 +x^13*y +x^12 -2*x^ 12*y^2 -x^11*y^3 -2*x^10*y^2 -x^10 -x^9*y^3 +x^8*y^4 -3*x^8 -3*x^8*y^2 -x^7*y -x^7*y^3 +x^6*y^4 +4*x^6*y^2 -y^3*x^5 +2*x^4 +y^2*x^4 -x^3*y +x^2 +x*y -1). - _R. J. Mathar_, May 02 2016
%F G.f. for column m=1: x +14*x^3 +2*x*(1 +2*x^2 +3*x^4 -2*x^6 -4*x^8 -2*x^10)/ (1-x^4-x^6)^2. - _R. J. Mathar_, May 02 2016, corrected Apr 10 2017
%F G.f. for column m=2: -8 +17*x^2 +2*x^4 -2*(9*x^16 +24*x^14 +17*x^12 -22*x^10 -39*x^8 -9*x^6 +13*x^4 +9*x^2 +4) / (x^6+x^4-1)^3. - _R. J. Mathar_, May 02 2016
%e The triangle starts in row n=1 and column m=0 as:
%e 0,3,0,4,0,1;
%e 6,0,35,0,26,0,1;
%e 0,18,0,56,0,16;
%e 3,0,52,0,64,0,7;
%e 0,10,0,88,0,80;
%e 2,0,60,0,182,0,81;
%e 0,8,0,160,0,320,0,96;
%e 2,0,102,0,500,0,449,0,112;
%e 0,18,0,340,0,952,0,600,0,120;
%e 4,0,184,0,1056,0,1535,0,712,0,128;
%e 0,24,0,550,0,2216,0,2338,0,824,0,128;
%e 4,0,246,0,2050,0,4367,0,3256,0,936,0,128;
%e 0,32,0,936,0,5044,0,7728,0,4454,0,1040,0,128;
%e 6,0,414,0,4054,0,11539,0,12360,0,5816,0,1160,0,128;
%e 0,52,0,1658,0,10736,0,22410,0,18744,0,7352,0,1280,0,128;
%e 8,0,620,0,7412,0,27039,0,39590,0,26576,0,9056,0,1408,0,128;
%e 0,68,0,2596,0,21180,0,57296,0,65634,0,36312,0,10864,0,1536,0,128;
%e 10,0,908,0,13022,0,59625,0,112526,0,102054,0,47954,0,12816,0,1664,0,128;
%e 0,100,0,4312,0,41056,0,138444,0,204496,0,152648,0,61720,0,14872,0,1792,0,128;
%e 14,0,1404,0,23112,0,126291,0,298136,0,347122,0,219228,0,77904,0,17056,0,1920,0,128;
%e 0,142,0,6904,0,77136,0,314464,0,584236,0,560856,0,305264,0,96552,0,19360,0,2048,0,128;
%e 18,0,2034,0,38898,0,254427,0,731536,0,1068766,0,863460,0,413418,0,117944,0,21792,0,2176,0,128;
%e 0,196,0,10778,0,139276,0,678728,0,1537620,0,1850598,0,1282412,0,546464,0,142128,0,24352,0,2304,0,128;
%e 24,0,3018,0,65388,0,496213,0,1704232,0,3026128,0,3048168,0,1843736,0,707754,0,169288,0,27040,0,2432,0,128;
%Y Cf. A192091 (row sums), A068924 (column m=0), A281791 (column m=1), A272473 (4 by n grid).
%K nonn,tabf
%O 1,2
%A _R. J. Mathar_, Apr 30 2016
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