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

3,3

Links

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

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)).

Example

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].

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 *)

Crossrefs

Sequence in context: A194280 A163362 A243061 * A112486 A342659 A253924

Adjacent sequences: A242908 A242909 A242910 * A242912 A242913 A242914

Keyword

nonn

Author

Alois P. Heinz, May 26 2014

Status

approved