A272473 Triangle T(n,m) by rows: the number of tatami tilings of a 4 by n grid with 2*m monomers.

1, 3, 1, 4, 18, 7, 4, 27, 13, 2, 32, 32, 3, 52, 64, 7, 3, 62, 133, 40, 3, 99, 269, 110, 9, 5, 152, 437, 280, 48, 5, 163, 730, 669, 138, 9, 6, 258, 1243, 1318, 433, 48, 8, 343, 1823, 2670, 1239, 154, 9, 8, 408, 2949, 5240, 2849, 600, 48, 11, 632, 4577

Offset: 1

Author

R. J. Mathar, Apr 30 2016

Keywords

• nonn
• tabf

Comments

The number of squares in the 4 by n floor is even, so the number of tilings with an odd number of monomers is zero.

Examples

			The triangle starts in row n=1 and column m=0 as:
1,3,1;
4,18,7;
4,27,13;
2,32,32;
3,52,64,7;
3,62,133,40;
3,99,269,110,9;
5,152,437,280,48;
5,163,730,669,138,9;
6,258,1243,1318,433,48;
8,343,1823,2670,1239,154,9;
8,408,2949,5240,2849,600,48;
11,632,4577,9011,6655,1927,172,9;
13,746,6287,16184,14697,4930,777,48;
14,971,9928,28135,28805,13089,2669,190,9;
19,1394,14234,44806,58022,32176,7501,954,48;
21,1610,19501,75702,111795,70427,22344,3445,208,9;
25,2224,29785,121302,199354,157078,59859,10576,1131,48;
32,2909,40073,184597,366553,331449,143611,34646,4257,226,9;
35,3464,55939,298278,644436,651772,350855,99300,14167,1308,48;
44,4820,81474,449995,1081033,1303651,802565,258303,50095,5105,244,9;
53,5924,106460,670726,1868914,2488996,1719501,684338,151835,18274,1485,48;
60,7408,150672,1040424,3077401,4548409,3716945,1678785,425017,68761,5989,262,9;
76,9972,208211,1503372,4956628,8434302,7641320,3879356,1208052,218806,22897,1662,48;
		

Crossrefs

Cf. A192090 (row sums), A068923 (column m=0), A272472 (3 by n grid), A100265 (without tatami condition, reversed rows).

Formula

G.f. x*( -1 -8*x^7*y^2 +21*x^5*y^2 -7*x^7*y^6 +4*x^3*y^2 -3*x^7 +2*x^5 -8*x^2*y^2 -4*x^8*y^4 -3*x -6*x*y^4 -15*x*y^2 -2*x^3*y^4 -6*x^8 -5*x^10*y^2 -5*x^9*y^2 -y^4 -2*x^8*y^2 -3*y^2 -8*x^11*y^2 +5*x^11*y^4 -3*x^2*y^4 -2*x^5*y^6 +2*x^13 +x^12 +x^11 +x^6 -7*x^7*y^4 +x^7*y^8 +11*x^4*y^2 -3*x^9 -15*x^10*y^4 -2*x^10*y^6 +18*x^9*y^4 +36*x^6*y^4 +20*x^6*y^2 -17*x^ 5*y^4 -8*x^4*y^4 +4*x^3 +8*x^6*y^6 +5*x^4 +2*x^9*y^6 -y^8*x^6 +6*y^6*x^3 +y^6*x^2)/ (x^11 -x^10 +2*x^9 -3*x^9*y^2 +x^8*y^2 -2*x^8 +x^7 +x^6*y^4 -5*x^6*y^2 -3*x^6 +2*x^5 +5*x^5*y^2 +x^4*y^2 -2*x^4 -x^3*y^2 +2*x^3 +x^2*y^2 +x -1). - R. J. Mathar, May 01 2016