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%I #37 Apr 12 2023 10:52:36
%S 1,1,1,1,2,2,2,5,9,9,4,13,34,64,64,9,35,119,326,625,625,20,95,401,
%T 1433,4016,7776,7776,48,262,1316,5799,21256,60387,117649,117649,115,
%U 727,4247,22224,100407,373895,1071904,2097152,2097152
%N 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.
%C T(n, k) where n counts the vertices and 0 <= k <= n counts the labels. - _Sean A. Irvine_, Mar 22 2018
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 134.
%H Sean A. Irvine, <a href="/A008295/b008295.txt">Table of n, a(n) for n = 0..527</a>
%H J. Riordan, <a href="https://dx.doi.org/10.1007/BF02392398">The numbers of labeled colored and chromatic trees</a>, Acta Mathematica 97 (1957) 211.
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F 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
%e Triangle begins with T(0,0):
%e n\k 0 1 2 3 4 5 6
%e 0 1
%e 1 1 1
%e 2 1 2 2
%e 3 2 5 9 9
%e 4 4 13 34 64 64
%e 5 9 35 119 326 625 625
%e 6 20 95 401 1433 4016 7776 7776
%t m = 9; r[_] = 0;
%t Do[r[x_] = x Exp[Sum[r[x^k]/k, {k, 1, j}]] + O[x]^j // Normal, {j, 1, m}];
%t r[x_, y_] = -ProductLog[(-E^(-r[x])) r[x] - (r[x] y)/E^r[x]];
%t (CoefficientList[#, y] Range[0, Exponent[#, y]]!)& /@ CoefficientList[r[x, y] + O[x]^m, x] /. {} -> {1} // Flatten // Quiet (* _Jean-François Alcover_, Oct 23 2019 *)
%Y Main diagonal is A000169.
%Y Columns k=0..6 are A000081, A000107, A000524, A000444, A000525, A064781, A064782.
%Y Cf. A034799.
%K nonn,tabl,nice
%O 0,5
%A _N. J. A. Sloane_
%E More terms from _Sean A. Irvine_, Mar 22 2018
%E Name edited by _Andrew Howroyd_, Mar 23 2023
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