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A385581 Square array read by antidiagonals: T(n,d) is the number of fixed d-dimensional polysticks of size n.

%I A385581 #8 Jul 06 2025 11:33:41
%S A385581 1,2,1,3,6,1,4,15,22,1,5,28,95,88,1,6,45,252,681,372,1,7,66,525,2600,
%T A385581 5277,1628,1,8,91,946,7065,29248,43086,7312,1,9,120,1547,15696,104097,
%U A385581 349132,365313,33466,1,10,153,2360,30513,285828,1632915,4351944,3186444,155446,1
%N A385581 Square array read by antidiagonals: T(n,d) is the number of fixed d-dimensional polysticks of size n.
%C A385581 The first 17 antidiagonals are from Mertens and Moore (2018), either directly from Table 1 or computed using the perimeter polynomials in Appendix A. T(14,5) is the only unknown value in the 18th antidiagonal.
%C A385581 T(13,6) = 14054816418877200 (Mertens and Moore).
%H A385581 Pontus von Brömssen, <a href="/A385581/b385581.txt">Table of n, a(n) for n = 1..153</a> (first 17 antidiagonals)
%H A385581 Stephan Mertens and Cristopher Moore, <a href="https://doi.org/10.1088/1751-8121/aae65c">Series expansion of the percolation threshold on hypercubic lattices</a>, J. Phys. A: Math. Theor., 51 (2018), 475001; arXiv:<a href="https://arxiv.org/abs/1805.02701">1805.02701</a> [cond-mat.stat-mech], 2018.
%H A385581 <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>.
%F A385581 T(n,d) = Sum_{k=1..d} binomial(n,k)*A385582(n,k) (with A385582(n,k) = 0 if d > n).
%e A385581 Table begins:
%e A385581   n\d| 1     2       3        4         5          6          7           8
%e A385581   ---+---------------------------------------------------------------------
%e A385581   1  | 1     2       3        4         5          6          7           8
%e A385581   2  | 1     6      15       28        45         66         91         120
%e A385581   3  | 1    22      95      252       525        946       1547        2360
%e A385581   4  | 1    88     681     2600      7065      15696      30513       53936
%e A385581   5  | 1   372    5277    29248    104097     285828     661549     1356384
%e A385581   6  | 1  1628   43086   349132   1632915    5551480   15314936    36449288
%e A385581   7  | 1  7312  365313  4351944  26817465  113045832  372033993  1028383408
%e A385581   8  | 1 33466 3186444 56062681 456137580 2386821009 9377038237 30118187174
%Y A385581 Cf. A000384 (row n=2), A385291 (polyominoes), A385582, A385583 (free).
%Y A385581 Columns: A096267 (d=2), A365560 (d=3), A365562 (d=4), A365564 (d=5).
%K A385581 nonn,tabl
%O A385581 1,2
%A A385581 _Pontus von Brömssen_, Jul 04 2025