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COMMENTS
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Successive rows:
p
0: 0, 0, 0, 0, 0, 0, 0, ...
1: 4, -3, 11, -8, 24, -19, 43, ...
2: 1, 8, 3, 16, 5, 24, 7, ...
3: 12, 0, 27, -3, 48, -12, 75, ...
4: 4, 24, 8, 40, 12, 56, 16, ...
5: 28, 5, 51, 4, 80, -3, 115, ...
6: 9, 48, 15, 72, 21, 96, 27, ...
... .
Main diagonal: alternate 3*n^2, -3.
From p>0, the rows are multiples of 1, 1, 3, 4, 1, 3, 1, 8, 3, 5, 1, 12, 1, 7, 3, 16, 1, ... . Sequences appearing after division: shifted A144433 or A195161, A064680. For p=3, we have (n+2)^2, -n^2.
First column: alternate n^2, 4*(n^2 + n + 1). Its first differences (4, -3, 11, -8, 24, ...) is the sequence of the square array for p=1.
Third column: 0, 3, 8, 15, ... is A005563(n).
Fifth column: 5, 21, 45, 77, ... is a bisection of A061037(n).
Seventh column: 7, 16, 40, 55, 91, 112, ... is a subsequence of A061039(n).
Etc. From the Rydberg spectra of the hydrogen atom (mentioned in A261327).
Starting for instance from p=-3,at the main antidiagonal,yields:
-3: -12, 0, -27, 3, ... see p=3
-2: -1, -8, -3, -16, -5, ... p=2
-1: -4, 3, -11, 8, -24, 19, ... p=1.
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