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A387002 Triangle read by rows: T(n,d) is the number of fixed, properly d-dimensional (d,2)-polyominoids of size n, 2 <= d <= n+1.

%I A387002 #13 Aug 19 2025 16:37:52
%S A387002 1,2,12,6,140,320,19,1554,10368,13520,63,17622,265344,892864,786432,
%T A387002 216,206747,6390484,41998840,89389920,58383808,760,2503578,152166240,
%U A387002 1749529040,6773387520
%N A387002 Triangle read by rows: T(n,d) is the number of fixed, properly d-dimensional (d,2)-polyominoids of size n, 2 <= d <= n+1.
%C A387002 A (d,2)-polyominoid consists of unit square cells with integer coordinates in the d-dimensional grid, where two cells are connected if they share an edge. The polyominoid is properly d-dimensional if it is not contained in a (d-1)-dimensional affine subspace.
%H A387002 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polyominoid">Polyominoid</a>.
%H A387002 <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>.
%F A387002 T(n,d) = Sum_{k=2..d} (-1)^(d-k)*binomial(d,k)*A385715(k,n), i.e., the n-th row is the inverse binomial transform of the n-th column of A385715 (with the convention that T(n,d) = A385715(d,n) = 0 when d <= 1).
%e A387002 Triangle begins:
%e A387002   n\d |     2          3           4           5          6        7  8  9 10 11
%e A387002   ----+-------------------------------------------------------------------------
%e A387002    1  |     1
%e A387002    2  |     2         12
%e A387002    3  |     6        140         320
%e A387002    4  |    19       1554       10368       13520
%e A387002    5  |    63      17622      265344      892864     786432
%e A387002    6  |   216     206747     6390484    41998840   89389920 58383808
%e A387002    7  |   760    2503578   152166240  1749529040 6773387520        ?  ?
%e A387002    8  |  2725   31117536  3644734836 69246650605          ?        ?  ?  ?
%e A387002    9  |  9910  394953243 88344741448           ?          ?        ?  ?  ?  ?
%e A387002   10  | 36446 5098388985           ?           ?          ?        ?  ?  ?  ?  ?
%Y A387002 Cf. A001168 (column d=2), A195739 (polyominoes), A385582 (polysticks), A385715, A387004 (free).
%K A387002 nonn,tabl,more,changed
%O A387002 1,2
%A A387002 _Pontus von Brömssen_, Aug 14 2025