A306318 Number of square twice-partitions of n.

1, 1, 1, 1, 2, 2, 4, 5, 10, 12, 19, 24, 39, 49, 73, 104, 151, 212, 317, 443, 638, 936, 1296, 1841, 2635, 3641, 5069, 7176, 9884, 13614, 19113, 26162, 36603, 50405, 70153, 96176, 135388, 184753, 257882, 353587, 494653, 671992, 934905, 1272195, 1762979, 2389255

Offset: 0

Author

Gus Wiseman, Feb 07 2019

Keywords

• nonn

Comments

A twice partition of n is a sequence of integer partitions, one of each part in an integer partition of n. It is square if the number of parts is equal to the number of parts in each part.

Examples

			The a(10) = 19 square twice-partitions:
  ((ten))  ((32)(32))  ((211)(111)(111))
           ((32)(41))
           ((33)(22))
           ((33)(31))
           ((41)(32))
           ((41)(41))
           ((42)(22))
           ((42)(31))
           ((43)(21))
           ((44)(11))
           ((51)(22))
           ((51)(31))
           ((52)(21))
           ((53)(11))
           ((61)(21))
           ((62)(11))
           ((71)(11))
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000219, A001970, A063834 (twice-partitions), A089299 (square plane partitions), A279787, A305551, A306017, A306319 (rectangular twice-partitions), A319066, A323429, A323531 (square partitions of partitions).

Programs

Mathematica

Table[Sum[Length[Union@@(Tuples[IntegerPartitions[#,{k}]&/@#]&/@IntegerPartitions[n,{k}])],{k,0,Sqrt[n]}],{n,0,20}]