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%I M4929 #75 Jun 28 2022 10:59:18
%S 0,14,42,90,165,275,429,637,910,1260,1700,2244,2907,3705,4655,5775,
%T 7084,8602,10350,12350,14625,17199,20097,23345,26970,31000,35464,
%U 40392,45815,51765,58275,65379,73112,81510,90610,100450,111069,122507,134805,148005
%N a(n) = n*(n+5)*(n+6)*(n+7)/24.
%C a(n) = number of Standard Young Tableaux of shape (n+3,4). - _David Callan_, Aug 17 2004
%C a(n) = A214292(n+6,3). - _Reinhard Zumkeller_, Jul 12 2012
%C a(n) for n > 0 is the number of n-extended coalescent histories for a matching caterpillar gene tree and species tree with 5 leaves. - _Noah A Rosenberg_, Jun 16 2022
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A005587/b005587.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Richard K. Guy, <a href="/A005581/a005581_1.pdf">Letter to N. J. A. Sloane, Feb 1988</a>.
%H Richard K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
%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 N. A. Rosenberg, <a href="https://doi.org/10.1089/cmb.2006.0109">Counting coalescent histories</a>, J. Comput. Biol. 14 (2007), 360-377.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F G.f.: (14 - 28*x + 20*x^2 - 5*x^3) / (1 - x)^5.
%F a(n) = C(7+n, 4) - C(7+n, 3). - _Zerinvary Lajos_, Dec 09 2005
%F E.g.f.: (1/24)*x*(336 + 168*x + 24*x^2 + x^3)*exp(x). - _G. C. Greubel_, Jul 01 2017
%F From _Amiram Eldar_, Jun 28 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 153/1225.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 288*log(2)/35 - 20759/3675. (End)
%p A005587:=z*(-14+28*z-20*z**2+5*z**3)/(z-1)**5; # _Simon Plouffe_ in his 1992 dissertation
%p seq(numbperm(n,4)/24-numbperm(n,3)/6, n=7..46); # _Zerinvary Lajos_, May 20 2008
%p a:=n->(sum(numbcomp(n,4), j=9..n)):seq(a(n)/4, n=8..47); # _Zerinvary Lajos_, Aug 26 2008
%t Table[n (n + 5) (n + 6) (n + 7)/24, {n, 0, 60}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 22 2011 *)
%t LinearRecurrence[{5,-10,10,-5,1},{0,14,42,90,165},40] (* _Harvey P. Dale_, Aug 17 2017 *)
%o (Magma) [n*(n+5)*(n+6)*(n+7)/24: n in [0..40]]; // _Vincenzo Librandi_, Mar 20 2013
%o (PARI) x='x+O('x^50); concat([0], Vec((14 - 28*x + 20*x^2 - 5*x^3) / (1 - x)^5)) \\ _G. C. Greubel_, Jul 01 2017
%Y Fifth diagonal of Catalan triangle A033184. Fifth column of Catalan triangle A009766.
%Y Numerator polynomial 14 - 28x + 20x^2 - 5x^3 from fourth row of triangle A062991.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E M4929 (this sequence) and M4930 were the same.
%E More terms from _Matthew Conroy_, Jan 16 2006
%E Plouffe Maple line edited by _N. J. A. Sloane_, May 13 2008
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