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A176042 For odd n, this is the number of partitions of n*(n-1)/2 with all part sizes between 3 and n, inclusive. For even n, this is the number of partitions of n*(n-2)/2 with all part sizes between 3 and n, inclusive. 1
1, 1, 2, 5, 20, 42, 238, 511, 3311, 7423, 52273, 119739, 894950, 2087761, 16317275, 38616848, 312598141, 748492526, 6233339701, 15070028915, 128475055100, 313137867019, 2722580871465, 6681890398543, 59076953846060, 145856049509351, 1308316471338448 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,3

COMMENTS

Number of different decompositions of complete graphs on n vertices into cycles (for odd n) or into cycles and a 1-factor (for n even) according to a conjecture of Brian Alspach's. This is verified for n <= 14.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 3..500

B. Alspach, Research problems, Problem 3, Discrete Math 36 (1981), page 333.

EXAMPLE

For n=3 we have 1+1+1 = 1+2 = 3 = n*(n-1)/2 of which only the final partition is counted.

For n=4 we have 1+1+1+1 = 1+1+2 = 2+2 = 1+3 = 4 = n*(n-2)/2 of which only the final partition is counted.

For n=5 we have n*(n-1)/2 = 10 and only 3+3+4 = 5+5 are counted.

MAPLE

a := proc (n) local i; option remember; if k::odd then coeff(series(1/(product(1-x^i, i = 3 .. n)), x, (1/2)*n*(n-1)+1), x, (1/2)*n*(n-1)) elif n::even then coeff(series(1/(product(1-x^i, i = 3 .. n)), x, (1/2)*n*(n-2)+1), x, (1/2)*n*(n-2)) end if end proc # Christopher Maitland, Jun 17 2010

# second Maple program:

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<3, 0,

      b(n, i-1)+b(n-i, min(n-i, i))))

    end:

a:= n-> `if`(n::odd, b(n*(n-1)/2, n), b(n*(n-2)/2, n)):

seq(a(n), n=3..35);  # Alois P. Heinz, Feb 21 2019

CROSSREFS

Sequence in context: A192164 A244087 A056726 * A185593 A136899 A136882

Adjacent sequences:  A176039 A176040 A176041 * A176043 A176044 A176045

KEYWORD

nonn

AUTHOR

Christopher Maitland (c3053540(AT)uon.edu.au), Apr 07 2010

EXTENSIONS

Definition edited by Franklin T. Adams-Watters, Nov 17 2011

STATUS

approved

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Last modified December 11 18:49 EST 2019. Contains 329925 sequences. (Running on oeis4.)