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A080242 Table of coefficients of polynomials P(n,x) defined by the relation P(n,x) = (1+x)P(n-1,x) + (-x)^(n+1). 4
1, 1, 1, 1, 1, 2, 2, 1, 3, 4, 2, 1, 1, 4, 7, 6, 3, 1, 5, 11, 13, 9, 3, 1, 1, 6, 16, 24, 22, 12, 4, 1, 7, 22, 40, 46, 34, 16, 4, 1, 1, 8, 29, 62, 86, 80, 50, 20, 5, 1, 9, 37, 91, 148, 166, 130, 70, 25, 5, 1, 1, 10, 46, 128, 239, 314, 296, 200, 95 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Values generate solutions to the recurrence a(n) = a(n-1) + k(k+1)a(n-2), a(0)=1, a(1) = k(k+1)+1. Values and sequences associated with this table are included in A072024.

LINKS

Table of n, a(n) for n=0..68.

FORMULA

Rows are generated by P(n, x)=((x+1)^(n+2)-(-x)^(n+2))/(2x+1)

The polynomials P(n,-x), n > 0, satisfy a Riemann hypothesis: their zeros lie on the vertical line Re x = 1/2 in the complex plane. O.g.f.: (1+x*t+x^2*t)/((1+x*t)(1-t-x*t)) = 1+(1+x+x^2)*t+(1+2x+2x^2)*t^2+ ... . - Peter Bala, Oct 24 2007

T(n,k)=if(k<=2*floor((n+1)/2), sum{j=0..floor((n+1)/2), binomial(n-2j,k-2j)},0). - Paul Barry, Apr 08 2011

EXAMPLE

Rows are {1}, {1,1,1}, {1,2,2}, {1,3,4,2,1}, {1,4,7,6,3}, ... This is the same as table A035317 with an extra 1 at the end of every second row.

Triangle begins

1,

1, 1, 1,

1, 2, 2,

1, 3, 4, 2, 1,

1, 4, 7, 6, 3,

1, 5, 11, 13, 9, 3, 1,

1, 6, 16, 24, 22, 12, 4,

1, 7, 22, 40, 46, 34, 16, 4, 1,

1, 8, 29, 62, 86, 80, 50, 20, 5

CROSSREFS

Similar to the triangles A059259, A035317, A108561, A112555. Cf. A059260.

Cf. A001045 (row sums).

Sequence in context: A076019 A071453 A212306 * A183927 A035317 A103923

Adjacent sequences:  A080239 A080240 A080241 * A080243 A080244 A080245

KEYWORD

easy,nonn,tabf

AUTHOR

Paul Barry, Feb 12 2003

STATUS

approved

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Last modified September 26 04:27 EDT 2017. Contains 292502 sequences.