A336167 Irregular triangular array read by rows. T(n,k) is the number of forests on n unlabeled nodes with exactly k distinct isomorphism classes of trees.

1, 0, 1, 0, 2, 0, 2, 1, 0, 4, 2, 0, 4, 6, 0, 9, 10, 1, 0, 12, 22, 3, 0, 27, 40, 9, 0, 49, 80, 24, 0, 111, 163, 53, 2, 0, 236, 342, 126, 6, 0, 562, 738, 280, 21, 0, 1302, 1662, 634, 60, 0, 3172, 3838, 1423, 165, 1, 0, 7746, 9041, 3308, 412, 7, 0, 19347, 21812, 7676, 1044, 26

Offset

0,5

Links

Table of n, a(n) for n=0..67.

Formula

O.g.f.: Product_{n>=1} (y/(1 - x^n) - y + 1)^A005195(n).

Example

1,
0, 1,
0, 2,
0, 2,   1,
0, 4,   2,
0, 4,   6,
0, 9,   10,  1,
0, 12,  22,  3,
0, 27,  40,  9,
0, 49,  80,  24,
0, 111, 163, 53,  2.

Mathematica

nn = 25; f[x_] := Sum[a[n] x^n, {n, 0, nn}]; sol =  SolveAlways[0 == Series[ f[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}], x]; r[x_] := Sum[a[n] x^n, {n, 0, nn}] /. sol; b = Drop[Flatten[CoefficientList[Series[r[x] - 1/2 (r[x]^2 - r[x^2]), {x, 0, nn}], x]], 1]; h[list_] := Prepend[Select[list, # > 0 &], 0];
Prepend[Drop[Map[h, CoefficientList[Series[Product[(y/(1 - x^k) - y + 1)^b[[k]], {k, 1, nn}], {x, 0, nn}], {x, y}]], 1], {1}] // Grid

Crossrefs

Cf. A035054 (column k=1), A005195 (row sums).

Sequence in context: A321428 A362615 A362614 * A076626 A182886 A108731

Adjacent sequences: A336164 A336165 A336166 * A336168 A336169 A336170

Keyword

nonn,tabf

Author

Geoffrey Critzer, Jul 10 2020

Status

approved