A294082 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.

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

Offset: 1

Author

David M. Cerna, Feb 09 2018

Keywords

• nonn
• easy
• tabl

Comments

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)) }.

Examples

			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
  ...
		

Crossrefs

Cf. A000058, A166105.

Programs

Mathematica

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 *)

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