Triangular + Stirling numbers

[Creighton Dement]

Dear Seqfans, 

One of my primary goals at the moment is to see what areas of math can
be "pulled closer together" with the help of floretions. Here are a few
results obtained by combining various rok-symmetries of
combinatorial/factorial type all with respect to a single floretion. The
user interface to FAMP has been updated to include the "roktypes"
necessary to generate the sequences, below. However, additional testing
is needed before I "officially" update FAMP on my websight. 

Sincerely, 
Creighton  
 

*************[Name: Multiply successively by 1,1,2,2,3,3,4,4,...]

http://www.crowdog.de/RokSym/A010551.html
http://www.research.att.com/projects/OEIS?Anum=A010551
Formula:   a(n) = [n/2]! * [(n+1)/2 ]! is the number of permutations p
of
              {1,2,3,...,n} such that for every i, i and p(i) have the
same parity,
              i.e. p(i) - i is even - Avi Peretz
(njk(AT)netvision.net.il), Feb 22
              2001
           a(n)=n!/binomial(n,floor(n/2)) - Paul Barry
(pbarry(AT)wit.ie),
              Sep 12 2004
           G.f: Sum_{n>=0} x^n/a(n) = besseli(0,2*x) + x*besseli(1,2*x).
-
              Paul D Hanna (pauldhanna(AT)juno.com), Apr 07 2005


**************[Name:  Number of permutations in the symmetric group S_n
that have odd order.]

http://www.crowdog.de/RokSym/A010551cycsigPermutationsofOddOrder.html

http://www.research.att.com/projects/OEIS?Anum=A000246


*************** [Name: Stirling numbers of first kind s(n,2):
a(n+1)=(n+1)*a(n)+n!.]

http://www.crowdog.de/RokSym/A010551StirlingNumbersofFirstKind.html


***************[Name:  n!*(n-1)!/2^(n-1).
Comments:  Product of first n triangular numbers. Might be called a
triangular factorial number. - Amarnath Murthy 
(amarnath_murthy(AT)yahoo.com), May 19 2002]

http://www.research.att.com/projects/OEIS?Anum=A006472
http://www.crowdog.de/RokSym/A010551newrokProductTriangularNumbers.html

http://www.crowdog.de/RokSym/A010551newrokProductTriangularNumberscycStirling.html


The next example is neat in that it is actually a force transform of the
same type as given for
http://www.research.att.com/projects/OEIS?Anum=A104770
(see "Generalized Sequence convergence?" link, a previous seqfan
message) - the link, below, gives the situation after the 10th
iteration.
*************[Determinant of n X n matrix whose diagonal are the first n
triangular
              numbers and all other elements are 1's.]
http://www.research.att.com/projects/OEIS?Anum=A067550
http://www.crowdog.de/RokSym/A010551TriangularNumbersDeterminant.html