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A080242
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Table of coefficients of polynomials P(n,x) defined by the relation P(n,x) = (1+x)P(n-1,x) + (-x)^(n+1).
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3
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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; internal format)
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OFFSET
| 0,6
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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 to this table are included in A072024
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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 (pbala(AT)toucansurf.com), 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, April 8 2011]
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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
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CROSSREFS
| Similar to the triangles A059259, A035317, A108561, A112555. Cf. A059260.
Cf. A001045 (row sums).
Sequence in context: A181512 A076019 A071453 * A183927 A035317 A103923
Adjacent sequences: A080239 A080240 A080241 * A080243 A080244 A080245
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KEYWORD
| easy,nonn,tabf
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Feb 12 2003
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