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Square array read by antidiagonals: T(m,n) = T(m,n-1)^2 - T(m,n-2)^2 + T(m,n-2) with T(1,n) = 1, T(m,0) = 1, and T(m,1) = m.
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%I #60 Mar 28 2018 14:18:53

%S 1,1,1,1,2,1,1,4,3,1,1,14,9,4,1,1,184,75,16,5,1,1,33674,5553,244,25,6,

%T 1,1,1133904604,30830259,59296,605,36,7,1,1,1285739649838492214,

%U 950504839176825,3515956324,365425,1266,49,8,1

%N Square array read by antidiagonals: T(m,n) = T(m,n-1)^2 - T(m,n-2)^2 + T(m,n-2) with T(1,n) = 1, T(m,0) = 1, and T(m,1) = m.

%C The columns of T(n,m) enumerate the size of the set S(n) constructed recursively as follows: Let S(1) = {a_1, ..., a_m}, where a_i are arbitrary elements, and let P(S) be the set of pairs (s,t) where s,t are members of S and s is not equal to t. We define S(n+1) to be the union of S(n) and P(S(n)). For example Let S(1) = {a_1,a_2}, then S(2) = {a_1,a_2, (a_1,a_2),(a_2,a_1)} where (a_1,a_2) is the pairing of a_{1} and a_{2}. Furthermore S(2) = {a_1,a_2, (a_1,a_2),(a_2,a_1), (a_1,(a_1,a_2)), (a_1,(a_2,a_1)), ((a_1,a_2),a_1), ((a_2,a_1),a_1), (a_2,(a_1,a_2)), (a_2,(a_2,a_1)), ((a_1,a_2),a_2), ((a_2,a_1),a_2)((a_1,a_2), (a_2,a_1)) }.

%e Array begins:

%e =============================================================================

%e m\n| 0 1 2 3 4 5 6

%e ---|-------------------------------------------------------------------------

%e 1 | 1 1 1 1 1 1 1

%e 2 | 1 2 4 14 184 33674 1133904604

%e 3 | 1 3 9 75 5553 30830259 950504839176825

%e 4 | 1 4 16 244 59296 3515956324 12361948868759636656

%e 5 | 1 5 25 605 365425 133535065205 17831613639170066626825

%e 6 | 1 6 36 1266 1601496 2564787836526 6578136646389154911912156

%e 7 | 1 7 49 2359 5562529 30941723313319 957390241597957573719482449

%e 8 | 1 8 64 4040 16317568 266263009117064 70895990024073440521846863040

%e ...

%t t[n_, m_] := t[n -1, m]^2 - t[n -2, m]^2 + t[n -2, m]; t[0, m_] := 1; t[1, m_] := m; Table[ t[n -m +1, m], {n, 0, 8}, {m, n +1}] // Flatten

%t (* to produce the table *) Table[t[n, m], {m, 8}, {n, 0, 6}] // TableForm (* _Robert G. Wilson v_, Feb 09 2018 *)

%o (PARI) T(n, k) = if (k<0, 0, if (n==1, 1, if (k==0, 1, if (k==1, n, T(n, k-1)^2 - T(n, k-2)^2 + T(n, k-2)))));

%o tabl(nn) = for (n=1, nn , for (k=0, nn, print1(T(n, k), ", ")); print); \\ _Michel Marcus_, Mar 06 2018

%Y Cf. A000058, A166105.

%K nonn,easy,tabl

%O 1,5

%A _David M. Cerna_, Feb 09 2018