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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. 0
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 (list; graph; refs; listen; history; text; internal format)
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: A157030 A080844 A321428 * A076626 A182886 A108731

Adjacent sequences:  A336164 A336165 A336166 * A336168 A336169 A336170

KEYWORD

nonn,tabf

AUTHOR

Geoffrey Critzer, Jul 10 2020

STATUS

approved

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Last modified June 16 11:58 EDT 2021. Contains 345057 sequences. (Running on oeis4.)