A277387 Number of nonnegative solutions of a certain system of linear Diophantine equations depending on an even parameter.
1, 17, 138, 670, 2355, 6671, 16212, 35148, 69765, 129085, 225566, 375882, 601783, 931035, 1398440, 2046936, 2928777, 4106793, 5655730, 7663670, 10233531, 13484647, 17554428, 22600100, 28800525, 36358101, 45500742, 56483938, 69592895, 85144755, 103490896, 125019312, 150157073, 179372865, 213179610
Offset: 1
Links
- Colin Barker, Table of n, a(n) for n = 1..1000
- Kamil Bradler, On the number of nonnegative solutions of a system of linear Diophantine equations, arXiv:1610.06387 [math-ph], 2016.
- Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
Crossrefs
Cf. A277388.
Programs
-
Mathematica
(* The code is in the InputForm form to simply copy and paste it in Mathematica. The input parameter is n>=0 (even) and for larger n's the code must be preceded by *) SetSystemOptions["ReduceOptions"->{"DiscreteSolutionBound"->1000}]; (* For a very large n the parameter value (1000) must be increased further but the enumeration is increasingly time-consuming. *) Reduce[Subscript[a,1,2]+Subscript[a,1,3]+Subscript[a,1,4]==n-2*Subscript[a,1,1]&&Subscript[a,1,2]>=0&&Subscript[a,1,3]>=0&&Subscript[a,1,4]>=0&&Subscript[a,1,1]>=0&&Subscript[a,1,2]+Subscript[a,2,3]+Subscript[a,2,4]==n-2*Subscript[a,2,2]&&Subscript[a,2,3]>=0&&Subscript[a,2,4]>=0&&Subscript[a,2,2]>=0&&Subscript[a,1,3]+Subscript[a,2,3]+Subscript[a,3,4]==n-2*Subscript[a,3,3]&&Subscript[a,3,4]>=0&&Subscript[a,3,3]>=0&&Subscript[a,1,4]+Subscript[a,2,4]+Subscript[a,3,4]==n-2*Subscript[a,4,4]&&Subscript[a,4,4]>=0,{Subscript[a,1,1],Subscript[a,1,2],Subscript[a,1,3],Subscript[a,1,4],Subscript[a,2,2],Subscript[a,2,3],Subscript[a,2,4],Subscript[a,3,3],Subscript[a,3,4],Subscript[a,4,4]},Integers]//Length (*For the special case n=0 the Reduce command must be put in the curly brackets before Length is applied.*) -
PARI
a(n) = (18+57*n+86*n^2+81*n^3+47*n^4+15*n^5+2*n^6)/18 \\ Colin Barker, Oct 12 2016
-
PARI
Vec(x*(1+10*x+40*x^2+26*x^3+3*x^4)/(1-x)^7 + O(x^30)) \\ Colin Barker, Oct 16 2016
Formula
a(n) = n*(1+n)*(3+2*n+n^2+n^3+2*n^4)/18.
From Colin Barker, Oct 12 2016: (Start)
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7) for n>7.
G.f.: x*(1+10*x+40*x^2+26*x^3+3*x^4) / (1-x)^7.
(End)
Comments