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

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, 1, 0, 5419, 2964, 1113, 328, 75, 12, 1, 0, 32816, 17890, 6792, 2078, 520, 102, 14, 1, 0, 203902, 110896, 42436, 13312, 3520, 772, 133, 16, 1

Offset: 0

Author

Peter Luschny, Oct 08 2022

Keywords

• nonn
• tabl

Comments

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

Examples

			Triangle T(n, k) starts:
[0] 1;
[1] 0,      1;
[2] 0,      2,      1;
[3] 0,      7,      4,     1;
[4] 0,     32,     18,     6,     1;
[5] 0,    166,     92,    33,     8,    1;
[6] 0,    926,    509,   188,    52,   10,  1;
[7] 0,   5419,   2964,  1113,   328,   75,  12,   1;
[8] 0,  32816,  17890,  6792,  2078,  520, 102,  14,  1;
[9] 0, 203902, 110896, 42436, 13312, 3520, 772, 133, 16, 1;
		

Crossrefs

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

Programs

Maple

InvPMatrix := proc(dim, seqfun) local k, m, M, A;
    if dim < 1 then return [] fi;
    A := [seq(seqfun(i), i = 1..dim-1)];
    M := Matrix(dim, shape=triangular[lower]); M[1, 1] := 1;
    for m from 2 to dim do
        M[m, m] := M[m - 1, m - 1] / A[1];
        for k from m-1 by -1 to 2 do
            M[m, k] := M[m - 1, k - 1] -
                add(A[i+1] * M[m, k + i], i = 1..m-k) / A[1]
od od; M end:
InvPMatrix(10, n -> [1, -2][irem(n-1, 2) + 1]);