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A101880
Number of arrangements of the partitions of n (e.g., 111 counts for 6).
11
1, 1, 3, 9, 35, 161, 913, 6103, 47319, 416235, 4092155, 44424095, 527511445, 6798907249, 94504286703, 1408973416617, 22426222745159, 379522092608177, 6804315177704869, 128828842646944135, 2568533750228603835, 53788282243854336411, 1180357162840624656959
OFFSET
0,3
COMMENTS
All terms are odd. - Alois P. Heinz, Jul 10 2023
LINKS
Jon Perry, Partition Tables [broken link]
FORMULA
a(n) = Sum_{i=1..n} P(n,i)*i!, where P(n,i) is the number of partitions of n into i parts.
G.f.: Sum_{n=1..infinity} (n!*x^n / Product_{k=1..n} (1-x^k)). - Vladeta Jovovic, Jan 29 2005
G.f.: ( 1 - G(0) )/(1-x) where G(k) = 1 - (k+1)/(1-x^(k+1))/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 22 2013
a(n) ~ n! * (1 + 1/n + 2/n^2 + 5/n^3 + 16/n^4 + 60/n^5 + 253/n^6 + 1180/n^7 + 6023/n^8 + 33306/n^9 + 197719/n^10 + ...), for coefficients see A331826. - Vaclav Kotesovec, Jan 28 2020
EXAMPLE
a(3) = 9 as we have 3, 12 (2) and 111 (6).
a(4) = 35 as 4, 31 (2), 22 (2), 211 (6) and 1111 (24).
MAPLE
b:= proc(n, i, p) option remember; `if`(n=0, p!,
`if`(i<1, 0, add(b(n-i*j, i-1, p+j), j=0..n/i)))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..25); # Alois P. Heinz, Apr 06 2016
MATHEMATICA
Rest[ CoefficientList[ Series[ Sum[ n!x^n / Product[1 - x^k, {k, n}], {n, 20}], {x, 0, 20}], x]] (* Robert G. Wilson v, Feb 10 2005 *)
PROG
(Sage)
from sage.combinat.partition import number_of_partitions_length
def A101880(n):
return sum(number_of_partitions_length(n, k)*factorial(k) for k in (0..n))
print([A101880(n) for n in (0..21)]) # Peter Luschny, Aug 01 2015
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Jon Perry, Jan 28 2005
EXTENSIONS
More terms from Vladeta Jovovic, Jan 29 2005
a(0)=1 prepended by Alois P. Heinz, Apr 06 2016
STATUS
approved