A204468 Number of 4-element subsets that can be chosen from {1,2,...,4*n} having element sum 8*n+2.

0, 1, 8, 33, 86, 177, 318, 519, 790, 1143, 1588, 2135, 2796, 3581, 4500, 5565, 6786, 8173, 9738, 11491, 13442, 15603, 17984, 20595, 23448, 26553, 29920, 33561, 37486, 41705, 46230, 51071, 56238, 61743, 67596, 73807, 80388, 87349, 94700, 102453, 110618, 119205

Offset: 0

Author

Alois P. Heinz, Jan 16 2012

Keywords

• nonn
• easy

Comments

a(n) is the number of partitions of 8*n+2 into 4 distinct parts <= 4*n.

Examples

			a(2) = 8 because there are 8 4-element subsets that can be chosen from {1,2,...,8} having element sum 18: {1,2,7,8}, {1,3,6,8}, {1,4,5,8}, {1,4,6,7}, {2,3,5,8}, {2,3,6,7}, {2,4,5,7}, {3,4,5,6}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3, -3, 2, -3, 3, -1).

Crossrefs

Column k=4 of A204459.

Programs

Maple

a:= n-> ((9+(16*n-18)*n)*n +[0, 2, -2][irem(n, 3)+1])/9:
seq(a(n), n=0..50);

Mathematica

LinearRecurrence[{3,-3,2,-3,3,-1},{0,1,8,33,86,177},50] (* or *) CoefficientList[Series[(x (1+5 x+12 x^2+9 x^3+5 x^4))/((-1+x)^4 (1+x+x^2)),{x,0,50}],x] (* Harvey P. Dale, Feb 25 2021 *)

Formula

G.f.: x*(5*x^4+9*x^3+12*x^2+5*x+1)/((x^2+x+1)*(x-1)^4).

a(n) = (1/9)*(2*A102283(n) + n*(16*n^2-18*n+9)). - Bruno Berselli, Jan 19 2012