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%I #46 Jun 18 2023 02:27:15
%S 1,15,85,295,771,1681,3235,5685,9325,14491,21561,30955,43135,58605,
%T 77911,101641,130425,164935,205885,254031,310171,375145,449835,535165,
%U 632101,741651,864865,1002835,1156695,1327621,1516831,1725585,1955185,2206975
%N Expansion of (1 + 10*x + 20*x^2 + 10*x^3 + x^4)/(1-x)^5.
%C Ehrhart series for matroid K_4.
%H Vincenzo Librandi, <a href="/A160747/b160747.txt">Table of n, a(n) for n = 0..10000</a>
%H P. Aluffi, <a href="https://arxiv.org/abs/1408.1702">Degrees of projections of rank loci</a>, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
%H Hyunsoo Cho, JiSun Huh, Hayan Nam, and Jaebum Sohn, <a href="https://arxiv.org/abs/2205.15554">Combinatorics on bounded free Motzkin paths and its applications</a>, arXiv:2205.15554 [math.CO], 2022. (See p. 14.)
%H J. A. De Loera, D. C. Haws and M. Koppe, <a href="http://arxiv.org/abs/0710.4346">Ehrhart Polynomials of Matroid Polytopes and Polymatroids</a>, arXiv:0710.4346 [math.CO], 2007; Discrete Comput. Geom., 42 (2009), 670-702.
%H D. C. Haws, <a href="http://www.math.ucdavis.edu/~haws/Matroids/">Matroids</a> [Broken link, Oct 30 2017]
%H D. C. Haws, <a href="https://www.math.ucdavis.edu/~mkoeppe/art/Matroids/">Matroids</a> [Copy on website of Matthias Koeppe]
%H D. C. Haws, <a href="/A160747/a160747.pdf">Matroids</a> [Cached copy, pdf file only]
%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.: (1 + 10*x + 20*x^2 + 10*x^3 + x^4)/(1-x)^5.
%F a(n) = 1 + 7*n*(n+1)*(n^2+n+2)/4. - _R. J. Mathar_, Dec 16 2009
%F E.g.f.: (1/4)*(7*x^4 + 56*x^3 + 112*x^2 + 56*x + 4)*exp(x). - _G. C. Greubel_, Apr 26 2018
%t Table[(7*n^4 + 14*n^3 + 21*n^2 + 14*n + 4)/4, {n,0,30}] (* _G. C. Greubel_, Apr 26 2018 *)
%o (Magma) [1+7*n*(n+1)*(n^2+n+2)/4: n in [0..40]]; // _Vincenzo Librandi_, Sep 18 2011
%o (PARI) a(n)=7*n*(n+1)*(n^2+n+2)/4+1 \\ _Charles R Greathouse IV_, Apr 17 2012
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Nov 18 2009
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