A274733 Number of odd partitions in the multiset of intersections of the set of partitions of n with itself three times; also number of distinct partitions in that multiset.

1, 1, 8, 26, 123, 334, 1295, 3222, 10172, 25300, 69258, 161259, 417582, 925972, 2200395, 4794092, 10769222, 22543912, 48728784, 98926942

Offset: 1

Author

George Beck, Jul 04 2016

Keywords

• nonn
• more

Comments

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

Form the p(n) x p(n) x p(n) matrix M of partitions of numbers ranging from 1 to n by taking the multiset intersections of all the triples 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 A260664. (conjectured)

Examples

			For an example for double intersections, see A274521.
		

Links

• George Beck, triple intersections of partitions.nb
• Sylvie Corteel, Carla D. Savage, Herbert S. Wilf, 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, A260664, A274521.