cp's OEIS Frontend

A381899 Irregular triangular array read by rows. T(n,k) is the number of length n words x on {0,1} such that I(x) + W(x)*(n-W(x)) = k, where I(x) is the number of inversions in x and W(x) is the number of 1's in x, n >= 0, 0 <= k <= floor(n^2/2).

%I A381899 #29 Mar 12 2025 15:55:04
%S A381899 1,2,2,1,1,2,0,2,2,2,2,0,0,2,3,3,4,1,1,2,0,0,0,2,2,4,4,6,4,4,2,2,2,0,
%T A381899 0,0,0,2,2,2,4,5,7,6,9,7,7,5,4,1,1,2,0,0,0,0,0,2,2,2,2,4,4,8,6,10,12,
%U A381899 14,12,14,10,10,6,4,2,2
%N A381899 Irregular triangular array read by rows. T(n,k) is the number of length n words x on {0,1} such that I(x) + W(x)*(n-W(x)) = k, where I(x) is the number of inversions in x and W(x) is the number of 1's in x, n >= 0, 0 <= k <= floor(n^2/2).
%C A381899 Sum_{k>=0} T(n,k)*2^k = A132186(n).
%C A381899 Sum_{k>=0} T(n,k)*3^k = A053846(n).
%C A381899 Sum_{k>=0} T(n,k)*q^k = the number of idempotent n X n matrices over GF(q).
%C A381899 It appears that if n is even the n-th row converges to 2,0,0,...,21,13,9,5,4,1,1 which is A226622 reversed, and if n is odd the sequence is twice A226635.
%C A381899 From _Alois P. Heinz_, Mar 09 2025: (Start)
%C A381899 Sum_{k>=0} k * T(n,k) = 3*A001788(n-1) for n>=1.
%C A381899 Sum_{k>=0} (-1)^k * T(n,k) = A060546(n). (End)
%H A381899 Alois P. Heinz, <a href="/A381899/b381899.txt">Rows n = 0..50, flattened</a>
%F A381899 Sum_{n>=0} Sum_{k>=0} T(n,k)*q^k*x^n/(n_q!*q^binomial(n,2)) = e(x)^2 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).
%e A381899 Triangle T(n,k) begins:
%e A381899   1;
%e A381899   2;
%e A381899   2, 1, 1;
%e A381899   2, 0, 2, 2, 2;
%e A381899   2, 0, 0, 2, 3, 3, 4, 1, 1;
%e A381899   2, 0, 0, 0, 2, 2, 4, 4, 6, 4, 4, 2, 2;
%e A381899   ...
%e A381899 T(4,5) = 3 because we have: {0, 1, 0, 0}, {0, 1, 0, 1}, {1, 1, 0, 1}.
%p A381899 b:= proc(i, j) option remember; expand(`if`(i+j=0, 1,
%p A381899      `if`(i=0, 0, b(i-1, j))+`if`(j=0, 0, b(i, j-1)*z^i)))
%p A381899     end:
%p A381899 T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(
%p A381899          expand(add(b(n-j, j)*z^(j*(n-j)), j=0..n))):
%p A381899 seq(T(n), n=0..10);  # _Alois P. Heinz_, Mar 09 2025
%t A381899 nn = 7; 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]^2, {z, 0, nn}],z]]
%Y A381899 Row sums give A000079.
%Y A381899 Cf. A001788, A060546, A226635, A226622, A132186, A053846, A083906.
%K A381899 nonn,tabf
%O A381899 0,2
%A A381899 _Geoffrey Critzer_, Mar 09 2025