David M. Cerna - cp's OEIS frontend

User: David M. Cerna

David M. Cerna has authored 2 sequences.

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

David M. Cerna, Feb 09 2018

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)) }.

			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
  ...

Cf. A000058, A166105.

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 *)

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

A244148 The number of ways one can assign values to n arrays a_{1},...,a_{n} of increasing size (size of a_{1} is 1, size of a_{2} is 2, ..., size of a_{n} is n) using the numbers 1, ..., n(n+1)/2, distinctly, such that the positions of array a_{i} can only be assigned values in the interval ((n+1)-i),... , (n(n+1)/2-(n-i)).

Original entry on oeis.org

1, 2, 72, 115200, 13276569600, 165253252792320000, 312379127174190543667200000, 120053472861445542607502662277529600000, 12098873398276702490569569159619238449643520000000000, 400639807706466477973460949403651522366500906696560470917120000000000

Offset: 1

David M. Cerna, Jun 21 2014

This sequence provides an upper bound for the following sequence: the number of ways one can assign values to n arrays a_{1},...,a_{n} of increasing size (size of a_{1} is 1, size of a_{2} is 2, ..., size of a_{n} is n) using the numbers 1, ..., n*(n+1)/2, distinctly, such that for the j^th position of array a_{i} (a_{i}(j)) one of the follow holds, a_{i+1}(j+1) < a_{i}(j) < a_{i+1}(j) or a_{i+1}(j) < a_{i}(j) < a_{i+1}(j+1). Currently, there is no formula known for enumerating this sequence.

David M. Cerna, Table of n, a(n) for n = 1..50
David M. Cerna, Proof of enumeration formula
Clark Kimberling, Unsolved Problems and Rewards: Number 18

a(n)=prod(k=1,n,k!* binomial((n^2 - 3*n + 5*k - k^2)/2 , k)); \\ Joerg Arndt, Jun 22 2014

a(n) = Prod_{k=1..n} (k!* binomial((n^2 - 3*n + 5*k - k^2)/2 , k)).

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User: David M. Cerna

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.

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