A320948 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).

2, 3, 7, 4, 13, 25, 4, 21, 58, 92, 6, 21, 113, 263, 345, 5, 43, 113, 614, 1203, 1311, 6, 31, 313, 614, 3351, 5531, 5030, 4, 43, 196, 2288, 3351, 18329, 25511, 19439, 6, 21, 313, 1247, 16749, 18329, 100372, 117910, 75545, 8, 43, 113, 2288, 7953, 122675, 100372, 550009, 545730, 294888

Offset

1,1

Links

Table of n, a(n) for n=1..55.

Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015 [table 5.1 p. 22].

Example

Array begins:
k:    2      3       4       5       6       7       8
n\p   3      5       7      11      13      17      19
1     2,     3,      4,      4,      6,      5,      6, ...
2     7,    13,     21,     21,     43,     31,     43, ...
3    25,    58,    113,    113,    313,    196,    313, ...
4    92,   263,    614,    614,   2288,   1247,   2288, ...
5   345,  1203,   3351,   3351,  16749,   7953,  16749, ...
6  1311,  5531,  18329,  18329, 122675,  50775, 122675, ...
7  5030, 25511, 100372, 100372, 898706, 324323, 898706, ...
...

Mathematica

rows = 10;
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 = Table[Array[om, rows] /. x -> x0, {x0, DivisorSigma[0, #-1]& /@ Prime[ Range[2, rows+1]]}] // Transpose;
Table[T[[n-k+2, k-1]], {n, 1, rows}, {k, n+1, 2, -1}] // Flatten

Crossrefs

Cf. A269750, A270785, A270786, A270787.

Sequence in context: A287950 A287628 A319863 * A086885 A324598 A229794

Adjacent sequences: A320945 A320946 A320947 * A320949 A320950 A320951

Keyword

nonn,tabl

Author

Jean-François Alcover, Oct 24 2018

Status

approved