OFFSET
0,2
COMMENTS
Values of Ehrhart polynomial for a facet of the Birkhoff polytope B_4.
LINKS
Jesús A. De Loera, Fu Liu, and Ruriko Yoshida, A generating function for all semi-magic squares and the volume of the Birkhoff polytope, J. Algebraic Combin. 30 (2009), no. 1, 113-139.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).
FORMULA
a(n) = (n+4)*(n+3)*(n+2)*(n+1)*(11*n^4+70*n^3+196*n^2+269*n+210)/5040.
G.f.: (1 + 9*x + 36*x^2 + 33*x^3 + 9*x^4)/(1 - x)^9. - Ilya Gutkovskiy, Apr 16 2016
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
MAPLE
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;
[seq(f(n), n=0..30)];
MATHEMATICA
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 *)
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 *)
PROG
(Python)
A271899_list, m = [], [88, -128, 61, -8]+[1]*5
for _ in range(100):
A271899_list.append(m[-1])
for i in range(8):
m[i+1] += m[i] # Chai Wah Wu, Apr 16 2016
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Apr 16 2016
STATUS
approved