A287428
Array read by antidiagonals: T(m,n) is the number of matchings in the stacked prism graph C_m X P_n.
Original entry on oeis.org
1, 2, 3, 3, 12, 4, 5, 47, 32, 7, 8, 185, 228, 108, 11, 13, 728, 1655, 1511, 342, 18, 21, 2865, 11978, 21497, 9213, 1104, 29, 34, 11275, 86731, 305184, 253880, 57536, 3544, 47, 55, 44372, 627960, 4334009, 6974078, 3079253, 356863, 11396, 76
Offset: 1
Table starts:
======================================================================
m\n| 1 2 3 4 5 6 7
---|------------------------------------------------------------------
1 | 1 2 3 5 8 13 21 ...
2 | 3 12 47 185 728 2865 11275 ...
3 | 4 32 228 1655 11978 86731 627960 ...
4 | 7 108 1511 21497 305184 4334009 61545775 ...
5 | 11 342 9213 253880 6974078 191668283 5267252351 ...
6 | 18 1104 57536 3079253 164206124 8761336545 467431319920 ...
7 | 29 3544 356863 37071837 3834744194 396924243197 41080815923665 ...
...
A102089
Triangle read by rows: T(n,k) is the number of k-matchings in the C_n X P_3 graph (C_n is the cycle graph on n vertices and P_3 is the path graph on 3 vertices).
Original entry on oeis.org
1, 10, 24, 12, 1, 15, 69, 107, 36, 1, 20, 142, 440, 588, 288, 32, 1, 25, 240, 1125, 2710, 3227, 1645, 240, 1, 30, 363, 2290, 8139, 16446, 18141, 9870, 2148, 108, 1, 35, 511, 4060, 19222, 55867, 99085, 103231, 58310, 15267, 1274, 1, 40, 684, 6560, 38934
Offset: 2
T(2,3)=12 because in the graph C_2 X P_3 with vertex set {A,B,C,A',B',C'} and edge set {AB,AC,A'B',B'C',a,a',b,b',c,c'}, where a and a' are two edges between A and A', b and b' are two edges between B and B' and c and c' are two edges between C and C', we have the following twelve3-matchings (as a matter of fact they are perfect matchings): eight 3-matchings by taking one edge from each of the pairs {a,a'},{b,b'} and {c,c'}; two 3-matchings by taking AB, A'B' and either edge from the pair {c,c'}; two 3-matchings by taking BC, B'C' and either edge from the pair {a,a'}.
Triangle starts:
1, 10, 24, 12;
1, 15, 69, 107, 36;
1, 20, 142, 440, 588, 288, 32;
1, 25, 240, 1125, 2710, 3227, 1645, 240;
- H. Hosoya and A. Motoyama, An effective algorithm for obtaining polynomials for dimer statistics. Application of operator technique on the topological index to two- and three-dimensional rectangular and torus lattices, J. Math. Physics 26 (1985) 157-167 (eq. (51) and Table VII).
-
G:= - z^2*( - 1 - 10*t + z^6*t^9 - 3*z^5*t^7 - 3*z^2*t^2 - 17*z^2*t^3 - z^3*t^3 + z^3*t^4 + 3*z^4*t^5 + 9*z^4*t^6 - 8*z^4*t^7 + 33*z^3*t^5 - 2*z^2*t^4 - 8*z^5*t^8 + t^12*z^7 - 4*t^8*z^4 + 49*t^6*z^3 + 48*t^5*z^2 - 3*t^9*z^5 - 4*t^11*z^6 - 36*t^9*z^4 + 40*t^7*z^3 + 40*t^6*z^2 - 26*t^10*z^5 + 2*z^7*t^13 + 8*t^12*z^6 - 25*z*t^2 - 47*z*t^3 - 12*z*t^4 - 3*z*t - 24*t^2 - 12*t^3)/(z^2*t^3 - 1 - z*t)/(z^6*t^9 - z^5*t^7 + z^5*t^6 - 5*z^4*t^6 - 3*z^4*t^5 - 2*z^4*t^4 - 2*z^3*t^4 + z^3*t^3 + 5*z^2*t^3 + z^3*t^2 + 7*z^2*t^2 + 2*z^2*t + 3*z*t + z - 1):
Gser:=simplify(series(G,z=0,13)): for n from 2 to 9 do P[n]:=coeff(Gser,z^n) od: b:=proc(n) if n mod 2 = 0 then 1 + 3*n/2 else 1 + b(n - 1) fi end:for n from 2 to 9 do seq(coeff(t*P[n],t^k),k=1..b(n)) od; # yields sequence in triangular form
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