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A294082
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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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0
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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, 1, 1, 1133904604, 30830259, 59296, 605, 36, 7, 1, 1, 1285739649838492214, 950504839176825, 3515956324, 365425, 1266, 49, 8, 1
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OFFSET
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1,5
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COMMENTS
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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)) }.
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LINKS
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EXAMPLE
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Array begins:
=============================================================================
m\n| 0 1 2 3 4 5 6
---|-------------------------------------------------------------------------
1 | 1 1 1 1 1 1 1
2 | 1 2 4 14 184 33674 1133904604
3 | 1 3 9 75 5553 30830259 950504839176825
4 | 1 4 16 244 59296 3515956324 12361948868759636656
5 | 1 5 25 605 365425 133535065205 17831613639170066626825
6 | 1 6 36 1266 1601496 2564787836526 6578136646389154911912156
7 | 1 7 49 2359 5562529 30941723313319 957390241597957573719482449
8 | 1 8 64 4040 16317568 266263009117064 70895990024073440521846863040
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MATHEMATICA
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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
(* to produce the table *) Table[t[n, m], {m, 8}, {n, 0, 6}] // TableForm (* Robert G. Wilson v, Feb 09 2018 *)
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PROG
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(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)))));
tabl(nn) = for (n=1, nn , for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, Mar 06 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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