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A000342 Number of n-node rooted trees of height 5.

%I A000342 M3884 N1594 #38 Jun 01 2022 14:44:37
%S A000342 0,0,0,0,0,1,5,19,61,180,498,1323,3405,8557,21103,51248,122898,291579,
%T A000342 685562,1599209,3705122,8532309,19543867,44552066,101124867,228640542,
%U A000342 515125815,1156829459,2590247002,5784031485,12883390590,28629914457
%N A000342 Number of n-node rooted trees of height 5.
%D A000342 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000342 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000342 N. J. A. Sloane, <a href="/A000342/b000342.txt">Table of n, a(n) for n=1..200</a>
%H A000342 J. Riordan, <a href="http://dx.doi.org/10.1147/rd.45.0473">Enumeration of trees by height and diameter</a>, IBM J. Res. Dev. 4 (1960), 473-478.
%H A000342 J. Riordan, <a href="/A007401/a007401_8.pdf">The enumeration of trees by height and diameter</a>, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
%H A000342 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H A000342 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A000342 A001385-A001384. (_Christian G. Bower_).
%p A000342 For Maple program see link in A000235.
%t A000342 f[n_] := Nest[CoefficientList[Series[Product[1/(1 - x^i)^#[[i]], {i, 1, Length[#]}], {x, 0, 40}], x] &, {1}, n];f[5]-f[4] (* _Geoffrey Critzer_, Aug 01 2013 *)
%t A000342 b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1 || k<1, 0, Sum[ Binomial[ b[i-1, i-1, k-1]+j-1, j]*b[n-i*j, i-1, k], {j, 0, n/i}]]]; a[n_] := b[n- 1, n-1, 5] - b[n-1, n-1, 4]; Array[a, 40] (* _Jean-François Alcover_, Feb 07 2016, after _Alois P. Heinz_ in A034781 *)
%Y A000342 Column h=5 of A034781.
%K A000342 nonn
%O A000342 1,7
%A A000342 _N. J. A. Sloane_