OFFSET
1,2
COMMENTS
Let aut(p) denote the size of the centralizer of the partition p (see A339016 for the definition). Then a(n) = Sum_{p in P} n!/aut(p), where P are the partitions of n with largest part k and length n + 1 - k. - Peter Luschny, Nov 19 2020
FORMULA
E.g.f.: exp(x)*(log(1/(1-x)) - x + 1). - Geoffrey Critzer, Nov 07 2015
EXAMPLE
A000522 begins 1 2 5 16 65 326 ...
with sums 1 3 8 24 89 415 ...
so sequence begins 1 2 6 21 85 410 ...
.
From Peter Luschny, Nov 19 2020: (Start):
The combinatorial interpretation is illustrated by this computation of a(5):
5! / aut([5]) = 120 / A339033(5, 1) = 120/5 = 24
5! / aut([4, 1]) = 120 / A339033(5, 2) = 120/4 = 30
5! / aut([3, 1, 1]) = 120 / A339033(5, 3) = 120/6 = 20
5! / aut([2, 1, 1, 1]) = 120 / A339033(5, 4) = 120/12 = 10
5! / aut([1, 1, 1, 1, 1]) = 120 / A339033(5, 5) = 120/120 = 1
--------------------------------------------------------------
Sum: a(5) = 85
(End)
MATHEMATICA
f[list_] :=Total[list]!/Apply[Times, list]/Apply[Times, Map[Length, Split[list]]!]; Table[Total[Map[f, Select[Partitions[n], Count[#, Except[1]] == 1 &]]] + 1, {n, 1, 20}] (* Geoffrey Critzer, Nov 07 2015 *)
PROG
(PARI) A000522(n)={ return( sum(k=0, n, n!/k!)) ; } A121726(n)={ return(sum(k=0, n-1, A000522(k))-n+1) ; } { for(n=1, 25, print1(A121726(n), ", ") ; ) ; } \\ R. J. Mathar, Sep 02 2006
(SageMath)
def A121726(n):
def h(n, k):
if n == k: return 1
return factorial(n)//((n + 1 - k)*factorial(k - 1))
return sum(h(n, k) for k in (1..n))
print([A121726(n) for n in (1..23)])
# Demonstrates the combinatorial view:
def A121726(n):
if n == 0: return 1
f = factorial(n); S = 0
for k in (0..n):
for p in Partitions(n, max_part=k, inner=[k], length=n+1-k):
S += (f // p.aut())
return S
print([A121726(n) for n in (1..23)]) # Peter Luschny, Nov 20 2020
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Alford Arnold, Aug 17 2006
EXTENSIONS
More terms from Franklin T. Adams-Watters, Aug 29 2006
More terms from R. J. Mathar, Sep 02 2006
STATUS
approved