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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 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.)