A325504 Product of products of parts over all strict integer partitions of n.

1, 1, 2, 6, 12, 120, 1440, 40320, 1209600, 1567641600, 2633637888000, 13905608048640000, 5046067048690483200000, 5289893008483207348224000000, 1266933607446134946465526579200000000, 99304891373531545064656621572980736000000000000

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..30

Formula

A001222(a(n)) = A325515(n).

a(n) = A003963(A325506(n)).

Example

The strict partitions of 5 are {(5), (4,1), (3,2)} with product a(5) = 5*4*1*3*2 = 120.
The sequence of terms together with their prime indices begins:
              1: {}
              1: {}
              2: {1}
              6: {1,2}
             12: {1,1,2}
            120: {1,1,1,2,3}
           1440: {1,1,1,1,1,2,2,3}
          40320: {1,1,1,1,1,1,1,2,2,3,4}
        1209600: {1,1,1,1,1,1,1,1,2,2,2,3,3,4}
     1567641600: {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4}
  2633637888000: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4}

Maple

a:= n-> mul(i, i=map(x-> x[], select(x->
        nops(x)=nops({x[]}), combinat[partition](n)))):
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 03 2021
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1$2], `if`(i<1, [0, 1], ((f, g)->
     [f[1]+g[1], f[2]*g[2]*i^g[1]])(b(n, i-1), b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 03 2021

Mathematica

Table[Times@@Join@@Select[IntegerPartitions[n], UnsameQ@@#&], {n, 0, 10}]

Crossrefs

Cf. A000009, A006128, A007870 (non-strict version), A015723, A022629 (sum of products of parts), A066186, A066189, A066633, A246867, A325505, A325506, A325512, A325513, A325515.

Sequence in context: A193987 A089423 A062349 * A193619 A290406 A371042

Adjacent sequences: A325501 A325502 A325503 * A325505 A325506 A325507

Keyword

nonn

Author

Gus Wiseman, May 07 2019

Status

approved