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A008295
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Triangle read by rows: T(n,k) is the number of partially labeled rooted trees with n vertices, k of which are labeled, 0 <= k <= n.
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9
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1, 1, 1, 1, 2, 2, 2, 5, 9, 9, 4, 13, 34, 64, 64, 9, 35, 119, 326, 625, 625, 20, 95, 401, 1433, 4016, 7776, 7776, 48, 262, 1316, 5799, 21256, 60387, 117649, 117649, 115, 727, 4247, 22224, 100407, 373895, 1071904, 2097152, 2097152
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OFFSET
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0,5
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COMMENTS
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T(n, k) where n counts the vertices and 0 <= k <= n counts the labels. - Sean A. Irvine, Mar 22 2018
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REFERENCES
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J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 134.
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LINKS
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FORMULA
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E.g.f.: r(x,y) = T(n,k) * y^k * x^n / k! satisfies r(x,y) * exp(r(x)) = (1+y) * r(x) * exp(r(x,y)) where r(x) is the o.g.f. for A000081. - Sean A. Irvine, Mar 22 2018
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EXAMPLE
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Triangle begins with T(0,0):
n\k 0 1 2 3 4 5 6
0 1
1 1 1
2 1 2 2
3 2 5 9 9
4 4 13 34 64 64
5 9 35 119 326 625 625
6 20 95 401 1433 4016 7776 7776
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MATHEMATICA
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m = 9; r[_] = 0;
Do[r[x_] = x Exp[Sum[r[x^k]/k, {k, 1, j}]] + O[x]^j // Normal, {j, 1, m}];
r[x_, y_] = -ProductLog[(-E^(-r[x])) r[x] - (r[x] y)/E^r[x]];
(CoefficientList[#, y] Range[0, Exponent[#, y]]!)& /@ CoefficientList[r[x, y] + O[x]^m, x] /. {} -> {1} // Flatten // Quiet (* Jean-François Alcover, Oct 23 2019 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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