OFFSET
0,3
COMMENTS
All terms are odd. - Alois P. Heinz, Jul 10 2023
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..450
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