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A334552 Array read by antidiagonals: T(m,n) is the number of fixed polyominoes that have a width of m and height of n and m + n - 1 cells.

%I A334552 #20 Dec 20 2023 16:30:00
%S A334552 1,1,1,1,4,1,1,8,8,1,1,12,25,12,1,1,16,50,50,16,1,1,20,83,120,83,20,1,
%T A334552 1,24,124,230,230,124,24,1,1,28,173,388,497,388,173,28,1,1,32,230,602,
%U A334552 932,932,602,230,32,1,1,36,295,880,1591,1924,1591,880,295,36,1
%N A334552 Array read by antidiagonals: T(m,n) is the number of fixed polyominoes that have a width of m and height of n and m + n - 1 cells.
%C A334552 A polyomino with a width of m and height of n must have at least m + n - 1 cells.
%H A334552 Andrew Howroyd, <a href="/A334552/b334552.txt">Table of n, a(n) for n = 1..1275</a>
%F A334552 T(m,n) = 2*binomial(m+n-2, m-1) + 2*(m+n-4) + (m-2)*(n-2)*(m+n-5) + 2*Sum_{i=1..m-2} Sum_{j=1..n-2} ((m-2-i)*(n-2-j)+2)*binomial(i+j,i) for m > 1, n > 1.
%F A334552 T(m,n) = max(1, 8*binomial(m+n-2, m-1) - 3*m*n + 2*m + 2*n - 8). - _Peter J. Taylor_, Dec 15 2020
%e A334552 Array begins:
%e A334552 =====================================================
%e A334552 m\n | 1  2   3    4    5     6     7     8      9
%e A334552 ----+------------------------------------------------
%e A334552   1 | 1  1   1    1    1     1     1     1      1 ...
%e A334552   2 | 1  4   8   12   16    20    24    28     32 ...
%e A334552   3 | 1  8  25   50   83   124   173   230    295 ...
%e A334552   4 | 1 12  50  120  230   388   602   880   1230 ...
%e A334552   5 | 1 16  83  230  497   932  1591  2538   3845 ...
%e A334552   6 | 1 20 124  388  932  1924  3588  6212  10156 ...
%e A334552   7 | 1 24 173  602 1591  3588  7265 13582  23859 ...
%e A334552   8 | 1 28 230  880 2538  6212 13582 27288  51290 ...
%e A334552   9 | 1 32 295 1230 3845 10156 23859 51290 102745 ...
%e A334552   ...
%t A334552 A334552[m_,n_]:=Max[1,8Binomial[m+n-2,m-1]-3m*n+2m+2n-8];
%t A334552 Table[A334552[m-n+1,n],{m,15},{n,m}] (* _Paolo Xausa_, Dec 20 2023 *)
%o A334552 (PARI) T(m, n)={if(m==1||n==1, 1, 8*binomial(m+n-2, m-1) - 3*m*n + 2*m + 2*n - 8)} \\ _Andrew Howroyd_, Dec 30 2020, after _Peter J. Taylor_
%Y A334552 Columns 2..3 are A008574(n-1), A164754(n+1).
%Y A334552 Main diagonal is A334551.
%Y A334552 Cf. A292357.
%K A334552 nonn,tabl
%O A334552 1,5
%A A334552 _Andrew Howroyd_, Jun 06 2020