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

%I A121726 #25 Sep 15 2024 03:35:30
%S A121726 1,2,6,21,85,410,2366,16065,125665,1112074,10976174,119481285,
%T A121726 1421542629,18348340114,255323504918,3809950976993,60683990530209,
%U A121726 1027542662934898,18430998766219318,349096664728623317,6962409983976703317,145841989688186383338,3201192743180799343822
%N A121726 Sum sequence A000522 then subtract 0,1,2,3,4,5,...
%C A121726 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
%F A121726 a(n) = A006231(n) + 1 = A002104(n) - (n-1). - _Franklin T. Adams-Watters_, Aug 29 2006
%F A121726 E.g.f.: exp(x)*(log(1/(1-x)) - x + 1). - _Geoffrey Critzer_, Nov 07 2015
%e A121726 A000522 begins     1 2 5 16 65 326 ...
%e A121726 with sums          1 3 8 24 89 415 ...
%e A121726 so sequence begins 1 2 6 21 85 410 ...
%e A121726 .
%e A121726 From _Peter Luschny_, Nov 19 2020: (Start):
%e A121726 The combinatorial interpretation is illustrated by this computation of a(5):
%e A121726 5! / aut([5])             = 120 / A339033(5, 1) = 120/5   = 24
%e A121726 5! / aut([4, 1])          = 120 / A339033(5, 2) = 120/4   = 30
%e A121726 5! / aut([3, 1, 1])       = 120 / A339033(5, 3) = 120/6   = 20
%e A121726 5! / aut([2, 1, 1, 1])    = 120 / A339033(5, 4) = 120/12  = 10
%e A121726 5! / aut([1, 1, 1, 1, 1]) = 120 / A339033(5, 5) = 120/120 =  1
%e A121726 --------------------------------------------------------------
%e A121726                                                 Sum: a(5) = 85
%e A121726 (End)
%t A121726 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 *)
%o A121726 (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
%o A121726 (SageMath)
%o A121726 def A121726(n):
%o A121726     def h(n, k):
%o A121726         if n == k: return 1
%o A121726         return factorial(n)//((n + 1 - k)*factorial(k - 1))
%o A121726     return sum(h(n, k) for k in (1..n))
%o A121726 print([A121726(n) for n in (1..23)])
%o A121726 # Demonstrates the combinatorial view:
%o A121726 def A121726(n):
%o A121726     if n == 0: return 1
%o A121726     f = factorial(n); S = 0
%o A121726     for k in (0..n):
%o A121726         for p in Partitions(n, max_part=k, inner=[k], length=n+1-k):
%o A121726             S += (f // p.aut())
%o A121726     return S
%o A121726 print([A121726(n) for n in (1..23)]) # _Peter Luschny_, Nov 20 2020
%Y A121726 Also the row sums of A092271.
%Y A121726 Cf. A000522, A006231, A002104, A339016, A339033.
%K A121726 easy,nonn
%O A121726 1,2
%A A121726 _Alford Arnold_, Aug 17 2006
%E A121726 More terms from _Franklin T. Adams-Watters_, Aug 29 2006
%E A121726 More terms from _R. J. Mathar_, Sep 02 2006