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%I #2 Mar 30 2012 18:59:45
%S 1,-1,1,-11,19,-9,1,-46,663,-3748,7711,-6606,2025,1,-130,6501,-163160,
%T 2236466,-17123340,71497186,-154127320,174334221,-98986050,22325625,1,
%U -295,36729,-2549775,109746165,-3080128275,57713313405,-727045264875
%N A sequence related to the recurrence relations of the right hand columns of the EG1 triangle A162005
%C 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).
%C 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].
%F 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.
%e The recurrence relations for the first few right hand columns:
%e n = 1: a(n) = 1*a(n-1)
%e n = 2: a(n) = 11*a(n-1)-19*a(n-2)+9*a(n-3)
%e 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)
%e 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)
%p 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);
%Y A000012, A004004 (2x), A162008, A162009 and A162010 are the first five right hand columns of EG1 triangle A162005.
%Y A000124 (the Lazy Caterer's sequence) gives the number of terms of the RR(n).
%Y A006324, A162012 and A162013 equal the absolute values of the coefficients that precede the a(n-1), a(n-2) and a(n-3) factors of the RR(n).
%K easy,sign,tabf
%O 1,4
%A _Johannes W. Meijer_, Jun 27 2009