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%I #41 Oct 06 2023 11:11:35
%S 1,18,162,927,3906,13248,38256,97614,225819,482482,965250,1827189,
%T 3299556,5720976,9574128,15531132,24508917,37735938,56831698,83900619,
%U 121641894,173477040,243696960,337630410,461835855,624318786,834776658,1104873697,1448547912,1882352736,2425835808
%N a(n) = 1 + 471*n/140 + 1594*n^2/315 + 73*n^3/16 + 161*n^4/60 + 83*n^5/80 + 61*n^6/240 + n^7/28 + 11*n^8/5040.
%C Values of Ehrhart polynomial for a facet of the Birkhoff polytope B_4.
%H Jesús A. De Loera, Fu Liu, and Ruriko Yoshida, <a href="https://www.emis.de/journals/JACO/Volume30_1/m6627810x2013373.html">A generating function for all semi-magic squares and the volume of the Birkhoff polytope</a>, J. Algebraic Combin. 30 (2009), no. 1, 113-139.
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F a(n) = (n+4)*(n+3)*(n+2)*(n+1)*(11*n^4+70*n^3+196*n^2+269*n+210)/5040.
%F G.f.: (1 + 9*x + 36*x^2 + 33*x^3 + 9*x^4)/(1 - x)^9. - _Ilya Gutkovskiy_, Apr 16 2016
%F a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9). - _Wesley Ivan Hurt_, Jul 02 2020
%p f:=n-> 1 + 471*n/140 + 1594*n^2/315 + 73*n^3/16 + 161*n^4/60 + 83*n^5/80 + 61*n^6/240 + n^7/28 + 11*n^8/5040;
%p [seq(f(n),n=0..30)];
%t CoefficientList[Series[(1 + 9 x + 36 x^2 + 33 x^3 + 9 x^4)/(1 - x)^9, {x, 0, 30}], x] (* _Michael De Vlieger_, Apr 16 2016 *)
%t LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,18,162,927,3906,13248,38256,97614,225819},40] (* _Harvey P. Dale_, Oct 06 2023 *)
%o (Python)
%o A271899_list, m = [], [88, -128, 61, -8]+[1]*5
%o for _ in range(100):
%o A271899_list.append(m[-1])
%o for i in range(8):
%o m[i+1] += m[i] # _Chai Wah Wu_, Apr 16 2016
%o (PARI) lista(nn) = for(n=0, nn, print1((1/5040)*(n+4)*(n+3)*(n+2)*(n+1)*(11*n^4+70*n^3+196*n^2+269*n+210), ", ")); \\ _Altug Alkan_, Apr 16 2016
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Apr 16 2016
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