A121726 Sum sequence A000522 then subtract 0,1,2,3,4,5,...

1, 2, 6, 21, 85, 410, 2366, 16065, 125665, 1112074, 10976174, 119481285, 1421542629, 18348340114, 255323504918, 3809950976993, 60683990530209, 1027542662934898, 18430998766219318, 349096664728623317, 6962409983976703317, 145841989688186383338, 3201192743180799343822

Offset: 1

Author

Alford Arnold, Aug 17 2006

Keywords

• easy
• nonn

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

Examples

			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)
		

Crossrefs

Also the row sums of A092271.

Cf. A000522, A006231, A002104, A339016, A339033.

Programs

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 *)

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

Formula

a(n) = A006231(n) + 1 = A002104(n) - (n-1). - Franklin T. Adams-Watters, Aug 29 2006

E.g.f.: exp(x)*(log(1/(1-x)) - x + 1). - Geoffrey Critzer, Nov 07 2015

Extensions

More terms from Franklin T. Adams-Watters, Aug 29 2006

More terms from R. J. Mathar, Sep 02 2006