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%I M4641 N1984 #40 Mar 24 2023 09:04:33
%S 9,64,326,1433,5799,22224,81987,293987,1031298,3555085,12081775,
%T 40576240,134919788,444805274,1455645411,4733022100,15302145060,
%U 49223709597,157629612076,502736717207,1597541346522,5059625685739,15975936032821,50304490599602
%N Number of partially labeled rooted trees with n nodes (3 of which are labeled).
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 134.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A000444/b000444.txt">Table of n, a(n) for n = 3..650</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F G.f.: A(x) = B(x)^3*(9-8*B(x)+2*B(x)^2)/(1-B(x))^5, where B(x) is g.f. for rooted trees with n nodes, cf. A000081.
%F a(n) ~ c * d^n * n^(3/2), where d = A051491 = 2.9557652856519949747148..., c = 0.244665117500618173509... . - _Vaclav Kotesovec_, Sep 11 2014
%p b:= proc(n) option remember; if n<=1 then n else add(k*b(k)* s(n-1, k), k=1..n-1)/(n-1) fi end: s:= proc(n,k) option remember; add(b(n+1-j*k), j=1..iquo(n,k)) end: B:= proc(n) option remember; add(b(k)*x^k, k=1..n) end: a:= n-> coeff(series(B(n-2)^3*(9-8*B(n-2)+2*B(n-2)^2)/(1-B(n-2))^5, x=0, n+1), x,n): seq(a(n), n=3..24); # _Alois P. Heinz_, Aug 21 2008
%t b[n_] := b[n] = If[n <= 1, n, Sum[k*b[k]*s[n-1, k], {k, 1, n-1}]/(n-1)]; s[n_, k_] := s[n, k] = Sum[b[n+1-j*k], {j, 1, Quotient[n, k]}]; B[n_] := B[n] = Sum [b[k]*x^k, {k, 1, n}]; a[n_] := Coefficient[Series[B[n-2]^3*(9 - 8*B[n-2] + 2*B[n-2]^2)/(1 - B[n-2])^5, {x, 0, n+1}], x, n]; Table[a[n], {n, 3, 30}] (* _Jean-François Alcover_, Mar 10 2014, after _Alois P. Heinz_ *)
%Y Column k=3 of A008295.
%Y Cf. A042977.
%K nonn
%O 3,1
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Oct 19 2001
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