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A357585 Triangle read by rows. Inverse of the convolution triangle of A108524, the number of ordered rooted trees with n generators.

%I A357585 #8 Oct 08 2022 07:44:07
%S A357585 1,0,1,0,2,1,0,7,4,1,0,32,18,6,1,0,166,92,33,8,1,0,926,509,188,52,10,
%T A357585 1,0,5419,2964,1113,328,75,12,1,0,32816,17890,6792,2078,520,102,14,1,
%U A357585 0,203902,110896,42436,13312,3520,772,133,16,1
%N A357585 Triangle read by rows. Inverse of the convolution triangle of A108524, the number of ordered rooted trees with n generators.
%C A357585 Also the matrix inverse of the signed version of A105475 with 1, 0, 0, 0, ... as column 0.
%e A357585 Triangle T(n, k) starts:
%e A357585 [0] 1;
%e A357585 [1] 0,      1;
%e A357585 [2] 0,      2,      1;
%e A357585 [3] 0,      7,      4,     1;
%e A357585 [4] 0,     32,     18,     6,     1;
%e A357585 [5] 0,    166,     92,    33,     8,    1;
%e A357585 [6] 0,    926,    509,   188,    52,   10,  1;
%e A357585 [7] 0,   5419,   2964,  1113,   328,   75,  12,   1;
%e A357585 [8] 0,  32816,  17890,  6792,  2078,  520, 102,  14,  1;
%e A357585 [9] 0, 203902, 110896, 42436, 13312, 3520, 772, 133, 16, 1;
%p A357585 InvPMatrix := proc(dim, seqfun) local k, m, M, A;
%p A357585     if dim < 1 then return [] fi;
%p A357585     A := [seq(seqfun(i), i = 1..dim-1)];
%p A357585     M := Matrix(dim, shape=triangular[lower]); M[1, 1] := 1;
%p A357585     for m from 2 to dim do
%p A357585         M[m, m] := M[m - 1, m - 1] / A[1];
%p A357585         for k from m-1 by -1 to 2 do
%p A357585             M[m, k] := M[m - 1, k - 1] -
%p A357585                 add(A[i+1] * M[m, k + i], i = 1..m-k) / A[1]
%p A357585 od od; M end:
%p A357585 InvPMatrix(10, n -> [1, -2][irem(n-1, 2) + 1]);
%Y A357585 Cf. A108524 (column 1), A047891 (row sums), A105475.
%K A357585 nonn,tabl
%O A357585 0,5
%A A357585 _Peter Luschny_, Oct 08 2022