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A242911 Half the number of compositions of n into exactly two different parts with equal multiplicities. 1
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Feb 28 2017, translated from Maple *)

CROSSREFS

Sequence in context: A194280 A163362 A243061 * A112486 A253924 A141410

Adjacent sequences:  A242908 A242909 A242910 * A242912 A242913 A242914

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 26 2014

STATUS

approved

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Last modified January 20 11:11 EST 2020. Contains 331083 sequences. (Running on oeis4.)