A242911 Half the number of compositions of n into exactly two different parts with equal multiplicities.

1, 1, 2, 5, 3, 6, 14, 10, 5, 56, 6, 15, 153, 51, 8, 502, 9, 217, 1756, 25, 11, 7023, 264, 30, 24363, 1852, 14, 93629, 15, 6576, 352782, 40, 3827, 1377543, 18, 45, 5200379, 105812, 20, 20063228, 21, 352942, 77607976, 55, 23, 301906830, 5172, 185320, 1166803215

Offset: 3

Author

Alois P. Heinz, May 26 2014

Keywords

• nonn

Examples

			a(6) = 5 because there are 10 compositions of 6 into exactly two different parts with equal multiplicities: [1,5], [5,1], [2,4], [4,2], [1,1,2,2], [1,2,1,2], [1,2,2,1], [2,1,1,2], [2,1,2,1], [2,2,1,1].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..1000

Programs

Maple

a:= n-> add(iquo(d-1, 2)*binomial(2*n/d, n/d),
        d=numtheory[divisors](n))/2:
seq(a(n), n=3..60);

Mathematica

a[n_] := DivisorSum[n, Quotient[#-1, 2]*Binomial[2n/#, n/#]&]/2; Table[ a[n], {n, 3, 60}] (* Jean-François Alcover, Feb 28 2017, translated from Maple *)

Formula

a(n) = 1/2 * Sum_{d|n} floor(d-1/2) * C(2*n/d,n/d).

a(p) = (p-1)/2 for odd prime p.

a(n) = 1/2 * (A131661(n)-A242900(n)).