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Triangle read by rows. Inverse of the convolution triangle of A108524, the number of ordered rooted trees with n generators.
2

%I #8 Oct 08 2022 07:44:07

%S 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 1,0,5419,2964,1113,328,75,12,1,0,32816,17890,6792,2078,520,102,14,1,

%U 0,203902,110896,42436,13312,3520,772,133,16,1

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

%C Also the matrix inverse of the signed version of A105475 with 1, 0, 0, 0, ... as column 0.

%e Triangle T(n, k) starts:

%e [0] 1;

%e [1] 0, 1;

%e [2] 0, 2, 1;

%e [3] 0, 7, 4, 1;

%e [4] 0, 32, 18, 6, 1;

%e [5] 0, 166, 92, 33, 8, 1;

%e [6] 0, 926, 509, 188, 52, 10, 1;

%e [7] 0, 5419, 2964, 1113, 328, 75, 12, 1;

%e [8] 0, 32816, 17890, 6792, 2078, 520, 102, 14, 1;

%e [9] 0, 203902, 110896, 42436, 13312, 3520, 772, 133, 16, 1;

%p InvPMatrix := proc(dim, seqfun) local k, m, M, A;

%p if dim < 1 then return [] fi;

%p A := [seq(seqfun(i), i = 1..dim-1)];

%p M := Matrix(dim, shape=triangular[lower]); M[1, 1] := 1;

%p for m from 2 to dim do

%p M[m, m] := M[m - 1, m - 1] / A[1];

%p for k from m-1 by -1 to 2 do

%p M[m, k] := M[m - 1, k - 1] -

%p add(A[i+1] * M[m, k + i], i = 1..m-k) / A[1]

%p od od; M end:

%p InvPMatrix(10, n -> [1, -2][irem(n-1, 2) + 1]);

%Y Cf. A108524 (column 1), A047891 (row sums), A105475.

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Oct 08 2022