Triangular + Stirling numbers
Creighton Dement
crowdog at crowdog.de
Tue Apr 12 15:31:07 CEST 2005
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 roksymmetries 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!*(n1)!/2^(n1).
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
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