This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A348622 #38 Jul 09 2025 04:56:56
%S A348622 1,1,1,1,6,6,1,28,168,168,1,120,3360,20160,20160,1,496,59520,1666560,
%T A348622 9999360,9999360,1,2016,999936,119992320,3359784960,20158709760,
%U A348622 20158709760,1,8128,16386048,8127479808,975297576960,27308332154880,163849992929280,163849992929280
%N A348622 Triangular array read by rows: T(n,k) is the number of periodic n X n matrices over GF(2) having rank k, n>=0, 0<=k<=n.
%C A348622 A matrix T is periodic if and only image(T) = image(T^2). Cf. A348015.
%H A348622 Alois P. Heinz, <a href="/A348622/b348622.txt">Rows n = 0..57, flattened</a>
%H A348622 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PeriodicMatrix.html">Periodic Matrix</a>
%F A348622 T(n,k) = A002884(n)/A002884(n-k).
%F A348622 T(n,k) = T(n,k-1)*T(n-k+1,1).
%F A348622 Sum_{n>=0} Sum_{k=0..n} T(n,k)*y^k*x^n/B(n) = e(x)*g(y*x) where e(x) = Sum_{n>=0} x^n/B(n), g(x) = Sum_{n>=0} Product_{i=0..n-1} (q^n-q^i)*x^n/B(n), B(n) = Product_{i=0..n-1} (q^n-q^i)/(q-1)^n and q=2. - _Geoffrey Critzer_, Jan 03 2025
%e A348622 Triangle begins:
%e A348622 1;
%e A348622 1, 1;
%e A348622 1, 6, 6;
%e A348622 1, 28, 168, 168;
%e A348622 1, 120, 3360, 20160, 20160;
%e A348622 1, 496, 59520, 1666560, 9999360, 9999360;
%e A348622 ...
%p A348622 b:= proc(n) option remember; mul(2^n-2^i, i=0..n-1) end:
%p A348622 T:= (n, k)-> b(n)/b(n-k):
%p A348622 seq(seq(T(n, k), k=0..n), n=0..8); # _Alois P. Heinz_, Oct 30 2021
%t A348622 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}]; l[greatestpart_]:=Level[Table[IntegerPartitions[n, {0, n}, Range[greatestpart]], {n, 0, nn}], {2}];
%t A348622 g1[u_, v_, deg_] :=Total[Map[v^(Length[#]) u^(deg Total[#])/aut[deg, #] &, l[1]]];
%t A348622 g2[u_, v_, deg_] := Total[Map[v^Length[#] u^(deg Total[#])/aut[deg, #] &,l[nn]]];
%t A348622 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
%Y A348622 Cf. A348015 (row sums).
%Y A348622 Main diagonal gives A002884.
%K A348622 nonn,tabl
%O A348622 0,5
%A A348622 _Geoffrey Critzer_, Oct 25 2021
%E A348622 Title improved by _Geoffrey Critzer_, Sep 16 2022