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A162011 A sequence related to the recurrence relations of the right hand columns of the EG1 triangle A162005 7
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 (list; graph; refs; listen; history; text; internal format)
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].

LINKS

Table of n, a(n) for n=1..32.

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).

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).

Sequence in context: A303237 A066950 A214495 * A123248 A111477 A238247

Adjacent sequences:  A162008 A162009 A162010 * A162012 A162013 A162014

KEYWORD

easy,sign,tabf

AUTHOR

Johannes W. Meijer, Jun 27 2009

STATUS

approved

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Last modified November 21 03:01 EST 2018. Contains 317427 sequences. (Running on oeis4.)