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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, 1, 8, 26, 123, 334, 1295, 3222, 10172, 25300, 69258, 161259, 417582, 925972, 2200395, 4794092, 10769222, 22543912, 48728784, 98926942 (list; graph; refs; listen; history; text; internal format)



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)


Table of n, a(n) for n=1..20.

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

H. S. Wilf, A pentagonal number sieve (with Sylvie Corteel, Carla Savage and Doron Zeilberger)


For an example for double intersections, see A274521.


Cf. A000009, A260664, A274521.

Sequence in context: A112645 A220713 A260962 * A223312 A194997 A089064

Adjacent sequences:  A274730 A274731 A274732 * A274734 A274735 A274736




George Beck, Jul 04 2016



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Last modified January 20 02:13 EST 2019. Contains 319320 sequences. (Running on oeis4.)