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Expansion of (1 + 80*x + 2592*x^2 + 29360*x^3 + 138124*x^4 + 295552*x^5 + 299984*x^6 + 144016*x^7 + 31146*x^8 + 2688*x^9 + 72*x^10)/(1-x)^16.
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%I #29 Sep 08 2022 08:46:16

%S 1,96,4008,82528,1029552,8939152,59112616,316345408,1429655844,

%T 5627681904,19747867728,62882889360,184259252180,502404837648,

%U 1286281062776,3115358788304,7182265229303,15843826126704,33591306176240,68708263470288,136027864081380

%N Expansion of (1 + 80*x + 2592*x^2 + 29360*x^3 + 138124*x^4 + 295552*x^5 + 299984*x^6 + 144016*x^7 + 31146*x^8 + 2688*x^9 + 72*x^10)/(1-x)^16.

%C Values of Ehrhart polynomial for a facet of the Birkhoff polytope B_5.

%C Linear recurrence signature is given by (-1)^n*binomial(16,n+1) for n=0..15. - _Bruno Berselli_, May 07 2016

%H Vincenzo Librandi, <a href="/A272765/b272765.txt">Table of n, a(n) for n = 0..1000</a>

%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 (see table on page 137, 3rd row).

%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).

%F G.f.: (1 + 80*x + 2592*x^2 + 29360*x^3 + 138124*x^4 + 295552*x^5 + 299984*x^6 + 144016*x^7 + 31146*x^8 + 2688*x^9 + 72*x^10)/(1 - x)^16.

%F a(n) = (1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(2179457280 + 5624791200*n + 8463690360*n^2 + 8430731628*n^3 + 5888536294*n^4 + 2935665243*n^5 + 1043021847*n^6 + 258468462 n^7 + 42566616*n^8 + 4198827*n^9 + 188723*n^10)/261534873600.

%F a(n) = 16*a(n-1) - 120*a(n-2) + 560*a(n-3) - 1820*a(n-4) + 4368*a(n-5) - 8008*a(n-6) + 11440*a(n-7) - 12870*a(n-8) + 11440*a(n-9) - 8008*a(n-10) + 4368*a(n-11) - 1820*a(n-12) + 560*a(n-13) - 120*a(n-14) + 16*a(n-15) - a(n-16). - _Wesley Ivan Hurt_, Jul 02 2020

%t CoefficientList[Series[(1 + 80 x + 2592 x^2 + 29360 x^3 + 138124 x^4 + 295552 x^5 + 299984 x^6 + 144016 x^7 + 31146 x^8 + 2688 x^9 + 72 x^10)/(1 - x)^16, {x, 0, 20}], x]

%o (PARI) Vec((1 + 80*x + 2592*x^2 + 29360*x^3 + 138124*x^4 + 295552*x^5 + 299984*x^6 + 144016*x^7 + 31146*x^8 + 2688*x^9 + 72*x^10)/(1 - x)^16 + O(x^21))

%o (Magma) m:=21; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 + 80*x + 2592*x^2 + 29360*x^3 + 138124*x^4 + 295552*x^5 + 299984*x^6 + 144016*x^7 + 31146*x^8 + 2688*x^9 + 72*x^10)/(1 - x)^16));

%Y Cf. A271899.

%K nonn,easy

%O 0,2

%A _Vincenzo Librandi_, May 06 2016

%E Edits, programs, new definition by _Bruno Berselli_, May 07 2016