A358833 Number of rectangular twice-partitions of n of type (P,R,P).

1, 1, 3, 4, 8, 8, 17, 16, 32, 34, 56, 57, 119, 102, 179, 199, 335, 298, 598, 491, 960, 925, 1441, 1256, 2966, 2026, 3726, 3800, 6488, 4566, 11726, 6843, 16176, 14109, 21824, 16688, 49507, 21638, 50286, 50394, 99408, 44584, 165129, 63262, 208853, 205109, 248150

Offset: 0

Author

Gus Wiseman, Dec 04 2022

Keywords

• nonn

Comments

A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n, so these are twice-partitions of n into partitions with constant lengths and constant sums.

Examples

			The a(1) = 1 through a(5) = 8 twice-partitions:
  (1)  (2)     (3)        (4)           (5)
       (11)    (21)       (22)          (32)
       (1)(1)  (111)      (31)          (41)
               (1)(1)(1)  (211)         (221)
                          (1111)        (311)
                          (2)(2)        (2111)
                          (11)(11)      (11111)
                          (1)(1)(1)(1)  (1)(1)(1)(1)(1)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• Gus Wiseman, Sequences enumerating triangles of integer partitions

Crossrefs

This is the rectangular case of A279787.

This is the case of A306319 with constant sums.

For distinct instead of constant lengths and sums we have A358832.

The version for multiset partitions of integer partitions is A358835.

A063834 counts twice-partitions, strict A296122, row-sums of A321449.

A281145 counts same-trees.

Cf. A000041, A000219, A001970, A008284, A141199, A327908, A358823, A358831.

Programs

Mathematica

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],SameQ@@Length/@#&&SameQ@@Total/@#&]],{n,0,10}]

PARI

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(u=Vec(P(n,y)-1)); concat([1], vector(n, n, sumdiv(n, d, my(p=u[n/d]); sum(j=1, n/d, polcoef(p, j, y)^d))))} \\ Andrew Howroyd, Dec 31 2022

Formula

a(n) = Sum_{d|n} Sum_{j=1..n/d} A008284(n/d, j)^d for n > 0. - Andrew Howroyd, Dec 31 2022

Extensions

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2022