A274521 Number of odd partitions in the multiset of intersections of the set of partitions of n with itself; also number of distinct partitions in that multiset.

1, 1, 4, 8, 23, 44, 107, 190, 406, 722, 1394, 2383, 4434, 7342, 12901, 21162, 35754, 57286, 94294, 147980, 237716, 368255, 577038, 880400, 1358074, 2043017, 3097194, 4607048, 6882358, 10121400, 14937754, 21726770, 31695300, 45685964, 65909693, 94165650

Offset: 1

Author

George Beck, Jun 26 2016

Keywords

• nonn

Comments

Let a(n) be the number of odd partitions in the multiset intersections of the set of partitions of n with itself.

Form the p(n) X p(n) matrix M of partitions of numbers ranging from 1 to n by taking the multiset intersections of all the pairs of partitions of n. Then, ignoring the empty set, the number of odd partitions in M equals the number of distinct partitions in M. (Proved in Wilf et al., "A pentagonal number sieve".)

By numerical experimentation, it seems a(n) is the convolution of A000009 (with offset 1) and A054440. (conjectured)

Examples

			For n=3, the partitions are 3, 21, 111. The multiset intersections are M = {{3, x, x}, {x, 21, 1}, {x, 1, 111}} (where x is the empty set), which fall into classes {{OD, y, y}, {y, D, OD}, {y, OD, O}}, where O means odd, D means distinct, OD means both, and y means neither. Thus a(3) = 4, the number of Os, which equals the number of Ds.
		

Links

• George Beck, Mathematica notebook
• Sylvie Corteel, Carla D. Savage, Herbert S. Wilf, and Doron Zeilberger, A pentagonal number sieve, J. Combin. Theory Ser. A 82 (1998), no. 2, 186-192.
• Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
• Wikipedia, Pentagonal number theorem

Crossrefs

Cf. A000009, A054440.