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Array T(n,k) of number of Schur rings over Z_{p^n} where n>=1 for p odd and k-th prime (by descending antidiagonals).
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%I #8 Oct 26 2018 09:26:42

%S 2,3,7,4,13,25,4,21,58,92,6,21,113,263,345,5,43,113,614,1203,1311,6,

%T 31,313,614,3351,5531,5030,4,43,196,2288,3351,18329,25511,19439,6,21,

%U 313,1247,16749,18329,100372,117910,75545,8,43,113,2288,7953,122675,100372,550009,545730,294888

%N Array T(n,k) of number of Schur rings over Z_{p^n} where n>=1 for p odd and k-th prime (by descending antidiagonals).

%H Andrew Misseldine, <a href="http://arxiv.org/abs/1508.03757">Counting Schur Rings over Cyclic Groups</a>, arXiv preprint arXiv:1508.03757 [math.RA], 2015 [table 5.1 p. 22].

%e Array begins:

%e k: 2 3 4 5 6 7 8

%e n\p 3 5 7 11 13 17 19

%e 1 2, 3, 4, 4, 6, 5, 6, ...

%e 2 7, 13, 21, 21, 43, 31, 43, ...

%e 3 25, 58, 113, 113, 313, 196, 313, ...

%e 4 92, 263, 614, 614, 2288, 1247, 2288, ...

%e 5 345, 1203, 3351, 3351, 16749, 7953, 16749, ...

%e 6 1311, 5531, 18329, 18329, 122675, 50775, 122675, ...

%e 7 5030, 25511, 100372, 100372, 898706, 324323, 898706, ...

%e ...

%t rows = 10;

%t om[n_] := om[n] = x om[n-1] + Sum[(CatalanNumber[k-1] x + 1) om[n - k], {k, 2, n}] // Expand; om[0] = 1; om[1] = x;

%t T = Table[Array[om, rows] /. x -> x0, {x0, DivisorSigma[0, #-1]& /@ Prime[ Range[2, rows+1]]}] // Transpose;

%t Table[T[[n-k+2, k-1]], {n, 1, rows}, {k, n+1, 2, -1}] // Flatten

%Y Cf. A269750, A270785, A270786, A270787.

%K nonn,tabl

%O 1,1

%A _Jean-François Alcover_, Oct 24 2018