Will do the edit(s), but probably later. * Bill Gosper <billgosper@gmail.com> [Oct 09. 2012 07:01]:
Suggested tweaks to the combinatorial interpretations of A003114: Also [number of partitions with all differences > 1, thus by Ferrers transposition,] number of partitions of n such that if k is the largest part, then each of {1, 2, ..., k-1} occur at least twice. Example: a(9)=5 because we have [3, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1].
(Schur's remark in http://mathworld.wolfram.com/Rogers-RamanujanIdentities.html. Caution: Identity (7) is not true for general a !) Also (http://en.wikipedia.org/wiki/Glaisher%27s_theorem).
Also (www.macalester.edu/~bressoud/pub/fpipc.tex) number of partitions into distinct parts with every even part > twice the number of odd parts. In[524]:= Table[ Length[Select[IntegerPartitions[k], FreeQ[Differences[#], 0] && Min[Select[#, EvenQ]] > 2*Length[Select[#, OddQ]] &]], {k, 0, 22}]
Out[524]= {1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, 31, 35, 41} --rwg
NJAS>
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