%I #32 Feb 18 2025 15:34:02
%S 1,1,4,1,3,4,1,6,6,4,1,5,22,12,4,1,8,30,82,24,4,1,7,86,160,306,48,4,1,
%T 10,126,776,850,1142,96,4,1,9,318,1484,7010,4520,4262,192,4,1,12,510,
%U 6114,18452,63674,24040,15906,384,4,1,11,1182,12348,126426,229698,578090,127860,59362,768,4
%N Array read by antidiagonals: T(n,k) is the number of Hamiltonian cycles in the stacked prism graph P_n X C_k, n >= 1, k >= 2.
%C The case for P_n X C_2 is determined using a double edge for C_2.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HamiltonianCycle.html">Hamiltonian Cycle</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StackedPrismGraph.html">Stacked Prism Graph</a>.
%e Array begins:
%e =========================================================
%e n\k | 2 3 4 5 6 7 8 ...
%e ----+---------------------------------------------------
%e 1 | 1 1 1 1 1 1 1 ...
%e 2 | 4 3 6 5 8 7 10 ...
%e 3 | 4 6 22 30 86 126 318 ...
%e 4 | 4 12 82 160 776 1484 6114 ...
%e 5 | 4 24 306 850 7010 18452 126426 ...
%e 6 | 4 48 1142 4520 63674 229698 2588218 ...
%e 7 | 4 96 4262 24040 578090 2861964 53055038 ...
%e 8 | 4 192 15906 127860 5247824 35663964 1087362018 ...
%e ...
%Y Columns 2..12 are A123932(n-1), A003945(n-1), A003699, A003731, A180582, A180583, A180584, A180585, A180586, A180587, A180588.
%Y Rows 1..2 are A000012, A103889(n+1).
%Y Cf. A222196 (order of recurrences), A222197 (main diagonal), A270273, A321172.
%K nonn,tabl,new
%O 1,3
%A _Andrew Howroyd_, Feb 18 2025