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A373396 Matrix inverse of triangle A296548, read by rows.

%I A373396 #14 Jun 07 2024 14:12:38
%S A373396 1,-1,1,5,-6,1,-113,140,-28,1,10879,-13560,2800,-120,1,-4324129,
%T A373396 5395984,-1120960,49600,-496,1,6984271295,-8717444064,1813050624,
%U A373396 -80709120,833280,-2016,1,-45479775838337,56768157085760,-11809230892032,526302695424,-5466697728,13655040,-8128,1
%N A373396 Matrix inverse of triangle A296548, read by rows.
%C A373396 Let P_n be the set of n X n idempotent matrices over GF(2) with the ordering:  E<=F iff EF=E=FE.  Then T(n,k) = Sum mu(0,E) where the sum is taken over the elements in P_n of rank k and mu is the Moebius mu incidence function of P_n.
%C A373396 To obtain the inverse, we regard the triangle as a lower triangular matrix, but then ignore the part above the diagonal.
%F A373396 Sum_{n>=0} Sum_{k=0..n} T(n,k)*y^k*x^n/A002884(n) = e(y*x)/e(x) where e(x) = Sum_{n>=0} x^n/A002884(n).
%e A373396  Triangle begins
%e A373396          1;
%e A373396         -1,       1;
%e A373396          5,      -6,        1;
%e A373396       -113,     140,      -28,     1;
%e A373396      10879,  -13560,     2800,  -120,    1;
%e A373396   -4324129, 5395984, -1120960, 49600, -496, 1;
%e A373396   ...
%t A373396 nn = 6; B[n_] = Product[q^n - q^i, {i, 0, n - 1}] /. q -> 2;e[x_] := Sum[x^n/B[n], {n, 0, nn}];Table[B[n], {n, 0, nn}]*CoefficientList[Series[ e[y x]/e[x], {x, 0, nn}], {x, y}] // Grid
%Y A373396 Cf. A296548, A002884.
%K A373396 sign,tabl
%O A373396 0,4
%A A373396 _Geoffrey Critzer_, Jun 03 2024