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A191662 a(n) = n! / A000034(n-1).

%I A191662 #36 Mar 28 2024 21:55:59
%S A191662 1,1,6,12,120,360,5040,20160,362880,1814400,39916800,239500800,
%T A191662 6227020800,43589145600,1307674368000,10461394944000,355687428096000,
%U A191662 3201186852864000,121645100408832000,1216451004088320000,51090942171709440000,562000363888803840000
%N A191662 a(n) = n! / A000034(n-1).
%C A191662 The a(n) are the denominators in the formulas of the k-dimensional square pyramidal numbers:
%C A191662 A005408 = (2*n+1)/1                       = 1, 3,  5,   7,   9, ... (k=1)
%C A191662 A000290 = (n^2)/1                         = 1, 4,  9,  16,  25, ... (k=2)
%C A191662 A000330 = n*(n+1)*(2*n+1)/6               = 1, 5, 14,  30,  55, ... (k=3)
%C A191662 A002415 = (n^2)*(n^2-1)/12                = 1, 6, 20,  50, 105, ... (k=4)
%C A191662 A005585 = n*(n+1)*(n+2)*(n+3)*(2*n+3)/120 = 1, 7, 27,  77, 182, ... (k=5)
%C A191662 A040977 = (n^2)*(n^2-1)*(n^2-4)/360       = 1, 8, 35, 112, 294, ... (k=6)
%C A191662 A050486 (k=7), A053347 (k=8), A054333 (k=9), A054334 (k=10), A057788 (k=11).
%C A191662 The first superdiagonal of this array appears in A029651. - _Paul Curtz_, Jul 04 2011
%C A191662 The general formula for the k-dimensional square pyramidal numbers is (2*n+k)*binomial(n+k-1,k-1)/k, k >= 1, n >= 0, see A097207. - _Johannes W. Meijer_, Jun 22 2011
%F A191662 a(2*n-1) = (2*n-1)!, a(2*n) = (2*n)!/2.
%F A191662 a(n+1) = A064680(n+1) * a(n).
%F A191662 From _Amiram Eldar_, Jul 06 2022: (Start)
%F A191662 Sum_{n>=1} 1/a(n) = sinh(1) + 2*cosh(1) - 2.
%F A191662 Sum_{n>=1} (-1)^(n+1)/a(n) = sinh(1) - 2*cosh(1) + 2. (End)
%F A191662 D-finite with recurrence: a(n) - (n-1)*n*a(n-2) = 0 for n >= 3 with a(1)=a(2)=1. - _Georg Fischer_, Nov 25 2022
%F A191662 a(n) = A052612(n)/2 for n >= 1. - _Alois P. Heinz_, Sep 05 2023
%p A191662 A191662:= proc(n): n!/A000034(n-1) end: A000034 := proc(n) op((n mod 2)+1, [1, 2]) ; end proc: seq(A191662(n),n=1..17); # _Johannes W. Meijer_, Jun 22 2011
%t A191662 Array[If[EvenQ[#],#!/2,#!]&,20] (* _Harvey P. Dale_, Mar 14 2014 *)
%Y A191662 Cf. A000142, A009445, A002674, A064680.
%Y A191662 Cf. A000290, A000330, A002415, A005408, A005585, A029651, A040977, A050486, A053347, A054333, A054334, A057788.
%Y A191662 Cf. A052612.
%K A191662 nonn,easy
%O A191662 1,3
%A A191662 _Paul Curtz_, Jun 10 2011
%E A191662 More terms from _Harvey P. Dale_, Mar 14 2014