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 A381930 #17 Mar 12 2025 15:58:48
%S A381930 1,3,3,3,3,3,0,6,7,8,2,1,3,0,0,6,9,12,18,12,12,6,3,3,0,0,0,6,6,12,15,
%T A381930 27,27,36,33,33,21,15,6,3,3,0,0,0,0,6,6,6,12,18,27,33,52,62,77,82,86,
%U A381930 75,68,48,35,19,11,2,1
%N A381930 Irregular triangular array read by rows. T(n,k) is the number of length n words x on {0,1,2} such that I(x) + W_0(x)*W_1(x) + W_0(x)*W_2(x) + W_1(x)*W_2(x) = k where I(x) is the number of inversions in x and W_i(x) is the number of occurrences of the letter i in x for i={0,1,2}, n>=0, 0<=k<=floor(2n^2/3).
%C A381930 Sum_{k>=0} T(n,k)*2^k = A342245(n).
%C A381930 Sum_{k>=0} T(n,k)*q^k = the number of ordered pairs (S,T) of idempotent n X n matrices over GF(q) such that ST=TS=S.
%H A381930 Alois P. Heinz, <a href="/A381930/b381930.txt">Rows n = 0..45, flattened</a>
%F A381930 Sum_{n>=0} Sum_{k>=0} T(n,k)*q^k*x^n/(n_q!*q^binomial(n,2)) = e(x)^3 where e(x) = Sum_{n>=0} x^n/(n_q!*q^binomial(n,2)) where n_q! = Product{i=1..n} (q^n-1)/(q-1).
%F A381930 From _Alois P. Heinz_, Mar 10 2025: (Start)
%F A381930 Sum_{k>=0} k * T(n,k) = 9 * A027472(n+1).
%F A381930 Sum_{k>=0} (-1)^k * T(n,k) = A056449(n). (End)
%e A381930 Triangle T(n,k) begins:
%e A381930 1;
%e A381930 3;
%e A381930 3, 3, 3;
%e A381930 3, 0, 6, 7, 8, 2, 1;
%e A381930 3, 0, 0, 6, 9, 12, 18, 12, 12, 6, 3;
%e A381930 3, 0, 0, 0, 6, 6, 12, 15, 27, 27, 36, 33, 33, 21, 15, 6, 3;
%e A381930 ...
%e A381930 T(3,3) = 7 because we have: {0, 1, 0}, {0, 1, 2}, {0, 2, 0}, {1, 0, 1}, {1, 2, 1}, {2, 0, 2}, {2, 1, 2}.
%p A381930 b:= proc(n, i, j, k) option remember; expand(
%p A381930 `if`(n=0, z^(i*j+i*k+j*k), b(n-1, i+1, j, k)*z^(j+k)+
%p A381930 b(n-1, i, j+1, k)*z^k +b(n-1, i, j, k+1)))
%p A381930 end:
%p A381930 T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$3)):
%p A381930 seq(T(n), n=0..10); # _Alois P. Heinz_, Mar 10 2025
%t A381930 nn = 6; B[n_] := FunctionExpand[QFactorial[n, q]]*q^Binomial[n, 2];e[z_] := Sum[z^n/B[n], {n, 0, nn}];Map[CoefficientList[#, q] &,Table[B[n], {n, 0, nn}] CoefficientList[Series[e[z]^3, {z, 0, nn}],z]] // Grid
%Y A381930 Cf. A000244 (row sums), A027472, A056449, A129529, A342245, A381899.
%K A381930 nonn,tabf
%O A381930 0,2
%A A381930 _Geoffrey Critzer_, Mar 10 2025