%I M4608 N1965 #50 May 15 2023 11:50:53
%S 9,30,69,133,230,369,560,814,1143,1560,2079,2715,3484,4403,5490,6764,
%T 8245,9954,11913,14145,16674,19525,22724,26298,30275,34684,39555,
%U 44919,50808,57255,64294,71960,80289,89318,99085,109629,120990,133209,146328,160390,175439,191520,208679
%N Powers of rooted tree enumerator.
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
%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 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%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>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = (n^4 + 18*n^3 + 83*n^2 + 114*n)/24. - _Philippe Deléham_, Feb 13 2004
%F G.f.: (2*x^3 - 9*x^2 + 15*x - 9)/(x - 1)^5. - _Jinyuan Wang_, Mar 17 2020
%p A000439:=(2*z-3)*(z**2-3*z+3)/(z-1)**5; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t Table[(n^4 + 18 n^3 + 83 n^2 + 114 n) / 24, {n, 50}] (* _Vincenzo Librandi_, Mar 18 2020 *)
%t LinearRecurrence[{5,-10,10,-5,1},{9,30,69,133,230},50] (* _Harvey P. Dale_, May 15 2023 *)
%o (PARI) a(n) = (n^4 + 18*n^3 + 83*n^2 + 114*n)/24; \\ _Jinyuan Wang_, Mar 17 2020
%o (Magma) [(n^4 + 18*n^3 + 83*n^2 + 114*n)/24: n in [1..50]]; // _Vincenzo Librandi_, Mar 18 2020
%o (Python)
%o def a(n): return (n**4 + 18*n**3 + 83*n**2 + 114*n)//24
%o print([a(n) for n in range(1, 44)]) # _Michael S. Branicky_, Sep 30 2021
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _Joerg Arndt_, May 09 2013