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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.
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%I #27 Nov 21 2020 16:22:45

%S 1,1,2,5,20,42,238,511,3311,7423,52273,119739,894950,2087761,16317275,

%T 38616848,312598141,748492526,6233339701,15070028915,128475055100,

%U 313137867019,2722580871465,6681890398543,59076953846060,145856049509351,1308316471338448

%N 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.

%C 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.

%H Alois P. Heinz, <a href="/A176042/b176042.txt">Table of n, a(n) for n = 3..500</a>

%H B. Alspach, <a href="https://doi.org/10.1016/S0012-365X(81)80029-5">Research problems, Problem 3</a>, Discrete Math 36 (1981), page 333.

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

%e 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.

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

%p 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

%p # second Maple program:

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

%p b(n, i-1)+b(n-i, min(n-i, i))))

%p end:

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

%p seq(a(n), n=3..35); # _Alois P. Heinz_, Feb 21 2019

%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 3, 0, b[n, i - 1] + b[n - i, Min[n - i, i]]]];

%t a[n_] := If[OddQ[n], b[n(n-1)/2, n], b[n(n-2)/2, n]];

%t a /@ Range[3, 35] (* _Jean-François Alcover_, Nov 21 2020, after _Alois P. Heinz_ *)

%K nonn

%O 3,3

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

%E Definition edited by _Franklin T. Adams-Watters_, Nov 17 2011