OFFSET
1,4
COMMENTS
The recurrence relation RR(n) = 0 of the n-th right hand column can be found with RR(n) = expand( product((1-(2*k-1)^2*z)^(n-k+1),k=1..n),z) = 0 and replacing z^p by a(n-p).
The polynomials in the numerators of the generating functions GF(z) of the coefficients that precede the a(n), a(n-1), a(n-2) and a(n-3) sequences, see A000012, A006324, A162012 and A162013, are symmetrical. This phenomenon leads to the sequence [1, 1, 6, 1, 19, 492, 1218, 492, 19 , 9, 3631, 115138, 718465, 1282314, 718465, 115138, 3631, 9].
FORMULA
RR(n) = expand( product((1-(2*k-1)^2*z)^(n-k+1),k=1..n),z) with n = 1, 2, 3, .. . The coefficients of these polynomials lead to the sequence given above.
EXAMPLE
The recurrence relations for the first few right hand columns:
n = 1: a(n) = 1*a(n-1)
n = 2: a(n) = 11*a(n-1)-19*a(n-2)+9*a(n-3)
n = 3: a(n) = 46*a(n-1)-663*a(n-2)+3748*a(n-3)-7711*a(n-4)+6606*a(n-5)-2025*a(n-6)
n = 4: a(n) = 130*a(n-1)-6501*a(n-2)+163160*a(n-3)-2236466*a(n-4)+17123340*a(n-5)-71497186*a(n-6)+154127320*a(n-7)-174334221*a(n-8)+98986050*a(n-9)-22325625*a(n-10)
MAPLE
nmax:=5; for n from 1 to nmax do RR(n) := expand(product((1-(2*k-1)^2*z)^(n-k+1), k=1..n), z) od: T:=1: for n from 1 to nmax do for m from 0 to(n)*(n+1)/2 do a(T):= coeff(RR(n), z, m): T:=T+1 od: od: seq(a(k), k=1..T-1);
CROSSREFS
A000012, A004004 (2x), A162008, A162009 and A162010 are the first five right hand columns of EG1 triangle A162005.
A000124 (the Lazy Caterer's sequence) gives the number of terms of the RR(n).
KEYWORD
easy,sign,tabf
AUTHOR
Johannes W. Meijer, Jun 27 2009
STATUS
approved