A230053 -id:A230053 - cp's OEIS frontend

A346788 Product over all partitions lambda of n of the product of distinct parts in lambda.

1, 1, 2, 6, 48, 1440, 414720, 2090188800, 1155790798848000, 226483217146419609600000, 302971317675145105975227187200000000, 37917003542135076706761224377027811868672000000000000, 45800346382799680410294841758069930049013501333211737122406400000000000000000

Offset: 0

Alois P. Heinz, Aug 03 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..20

Cf. A000142, A007870, A162506, A230053, A325504.

a:= n-> mul(i, i=map(x-> {x[]}[], combinat[partition](n))):
seq(a(n), n=0..12);

a[n_] := Times @@ Times @@@ Union /@ IntegerPartitions[n];
a /@ Range[0, 20] (* Jean-François Alcover, Aug 09 2021 *)

a(n) = vecprod(apply(x->vecprod(Set(x)), partitions(n))); \\ Michel Marcus, Aug 04 2021

a(n) = A230053(n) for n <= 6.

A380023 a(n) = a(n-1) * a(n-2) * (1 - 2 / (n * (n-1))), with a(1) = 2, a(2) = 3.

Original entry on oeis.org

2, 3, 4, 10, 36, 336, 11520, 3732480, 41803776000, 152564385447936000, 6261807987664209366220800000, 940854207318376503485146088437972992000000000, 5815917000990435607656487842294594291938222391518950745702400000000000000

Offset: 1

Hamza K. Özer, Jan 09 2025

nonn

This sequence is produced when Moessner's Magic is applied to the Fibonacci sequence.

			For n = 6, a(6) = 336 solutions are 36 * 10 * (1 - 2/(6*5)) = 336 or 7 * 5^0 * 4^1 * 3^1 * 2^2 = 336.

J. H. Conway and R. K. Guy, "The Book of Numbers", Springer-Verlag, 1996, page 63

Wikipedia, Moessner's Theorem

Cf. A000045, A230053.

RecurrenceTable[{a[n] == a[n-1]*a[n-2]*(1 - 2/(n*(n-1))), a[1] == 2, a[2] == 3}, a, {n, 1, 14}] (* Amiram Eldar, Jan 09 2025 *)

import math
def A380023(n): return None if n < 1 else (n + 1) * math.prod([(n - k - 1) ** fibo(k) for k in range(0, n - 1)])
# fibo(k) is the function for calculating the k-th Fibonacci number

a(n) = (n+1) * A230053(n-2), for n >= 2.
a(n) = (n+1) * Product_{k=0..n-2} (n-k-1)^Fibonacci(k), for n >= 2.
Using Moessner's Magic (page 63 of The Book of Numbers), start with writing the counting numbers and cross out each Fibonacci number. Underneath, write the partial sums of the uncrossed terms in the first row and cross out the terms which are offset one place to left of the crossed out terms in the first row. Continuing this process for successive rows produces this sequence:
1 2 3 4 5 6  7  8 9  10  11  12 13
* * * 4 * 10 17 * 26 36  45  57 *
* * * * * 10 *  * 36 72  117 *  *
* * * * * *  *  * 36 108 *   *  *
* * * * * *  *  * 36 *   *   *  *

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A346788 Product over all partitions lambda of n of the product of distinct parts in lambda.

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A380023 a(n) = a(n-1) * a(n-2) * (1 - 2 / (n * (n-1))), with a(1) = 2, a(2) = 3.

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