A274521 - OEIS

%I #33 Feb 16 2025 08:33:36

%S 1,1,4,8,23,44,107,190,406,722,1394,2383,4434,7342,12901,21162,35754,

%T 57286,94294,147980,237716,368255,577038,880400,1358074,2043017,

%U 3097194,4607048,6882358,10121400,14937754,21726770,31695300,45685964,65909693,94165650

%N 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.

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

%C 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".)

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

%H George Beck, <a href="/A274521/a274521.nb">Mathematica notebook</a>

%H Sylvie Corteel, Carla D. Savage, Herbert S. Wilf, and Doron Zeilberger, <a href="http://dx.doi.org/10.1006/jcta.1997.2846">A pentagonal number sieve</a>, J. Combin. Theory Ser. A 82 (1998), no. 2, 186-192.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalNumberTheorem.html">Pentagonal Number Theorem</a>

%H Wikipedia, <a href="http://www.wikipedia.org/wiki/Pentagonal_number_theorem">Pentagonal number theorem</a>

%e 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.

%Y Cf. A000009, A054440.

%K nonn

%O 1,3

%A _George Beck_, Jun 26 2016