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A162011
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A sequence related to the recurrence relations of the right hand columns of the EG1 triangle A162005
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7
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1, -1, 1, -11, 19, -9, 1, -46, 663, -3748, 7711, -6606, 2025, 1, -130, 6501, -163160, 2236466, -17123340, 71497186, -154127320, 174334221, -98986050, 22325625, 1, -295, 36729, -2549775, 109746165, -3080128275, 57713313405, -727045264875
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OFFSET
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1,4
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COMMENTS
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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].
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LINKS
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FORMULA
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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.
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EXAMPLE
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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)
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MAPLE
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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);
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CROSSREFS
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A000124 (the Lazy Caterer's sequence) gives the number of terms of the RR(n).
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).
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KEYWORD
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easy,sign,tabf
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AUTHOR
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STATUS
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approved
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