OFFSET
0,5
COMMENTS
A matrix T is periodic if and only image(T) = image(T^2). Cf. A348015.
LINKS
Alois P. Heinz, Rows n = 0..57, flattened
Eric Weisstein's World of Mathematics, Periodic Matrix
EXAMPLE
Triangle begins:
1;
1, 1;
1, 6, 6;
1, 28, 168, 168;
1, 120, 3360, 20160, 20160;
1, 496, 59520, 1666560, 9999360, 9999360;
...
MAPLE
b:= proc(n) option remember; mul(2^n-2^i, i=0..n-1) end:
T:= (n, k)-> b(n)/b(n-k):
seq(seq(T(n, k), k=0..n), n=0..8); # Alois P. Heinz, Oct 30 2021
MATHEMATICA
nn = 7; q = 2; b[p_, i_] := Count[p, i]; s[p_, i_] := Sum[j b[p, j], {j, 1, i}] + i Sum[b[p, j], {j, i + 1, Total[p]}]; aut[deg_, p_] := Product[Product[q^(s[p, i] deg) - q^((s[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1, Total[p]}]; \[Nu] = Table[1/n Sum[MoebiusMu[n/m] q^m, {m, Divisors[n]}], {n, 1, nn}]; Lt = Prepend[Table[{n}, {n, 1, nn}], }]; l[greatestpart_]:=Level[Table[IntegerPartitions[n, {0, n}, Range[greatestpart]], {n, 0, nn}], {2}];
g1[u_, v_, deg_] :=Total[Map[v^(Length[#]) u^(deg Total[#])/aut[deg, #] &, l[1]]];
g2[u_, v_, deg_] := Total[Map[v^Length[#] u^(deg Total[#])/aut[deg, #] &, l[nn]]];
Map[Reverse, Map[Select[#, # > 0 &] &, Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[g1[u, v, 1] g2[u, 1, 1]^(q - 1) Product[g2[u, 1, d]^\[Nu][[d]], {d, 2, nn}] , {u, 0, nn}], {u, v}]]] // Flatten
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Oct 25 2021.
EXTENSIONS
Title improved by Geoffrey Critzer, Sep 16 2022
STATUS
approved