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A135929
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Triangle read by rows: row n gives coefficients of polynomial P_n(x)= U_n(X,1) + 3 * U_{n-2}(X,1) where U is the Chebyshev polynomial of the second kind, in order of decreasing exponents
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19
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1, 1, 0, 1, 0, 2, 1, 0, 1, 0, 1, 0, 0, 0, -2, 1, 0, -1, 0, -3, 0, 1, 0, -2, 0, -3, 0, 2, 1, 0, -3, 0, -2, 0, 5, 0, 1, 0, -4, 0, 0, 0, 8, 0, -2, 1, 0, -5, 0, 3, 0, 10, 0, -7, 0, 1, 0, -6, 0, 7, 0, 10, 0, -15, 0, 2, 1, 0, -7, 0, 12, 0, 7, 0, -25, 0, 9, 0, 1, 0, -8, 0, 18, 0, 0, 0, -35, 0, 24, 0, -2, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| Take a(0)=-2 instead of 1. The recurrence begins immediately (at the third instead of the fourth polynomial). Companion: A192011(n). - Paul Curtz, Sep 20 2011
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REFERENCES
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972; see Chapter 22.
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FORMULA
| G.f.: (1+3*t^2)/(1-x*t+t^2).
P_n(X) = U_n(X,1) + 3 * U_{n-2}(X,1) for n>=2 [Max Alekseyev, Dec 04 2009]
P_n(x) = S_n(x)+3*S_{n-2}(x), with Chebyshev Polynomials S_n(x) defined in A049310 and A053119. [R. J. Mathar, Dec 07 2009]
P_0(x)=1, P_1(x)=x, P_2(x)=x^2+2, and P_n(x)= x*P_{n-1}(x) - P_{n-2}(x) for n>=3. [Paul Curtz, Aug 14 2011]
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EXAMPLE
| The polynomials are
1; 1
1,0; x
1,0,2; x^2+2
1,0,1,0; x^3+x
1,0,0,0,-2; x^4-2
1,0,-1,0,-3,0; x^5-x^3-3*x
1,0,-2,0,-3,0,2; x^6-2*x^4-3*x^2+2
1,0,-3,0,-2,0,5,0; x^7-3*x^5-2*x^3+5*x
1,0,-4,0,0,0,8,0,-2; x^8-4*x^6+8*x^2-2
1,0,-5,0,3,0,10,0,-7,0; x^9-5*x^7+3*x^5+10*x^3-7*x
...
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MAPLE
| A135929 := proc(n, m) coeftayl( coeftayl( (1+3*t^2)/(1-x*t+t^2), t=0, n), x=0, n-m) ; end proc: seq(seq(A135929(n, m), m=0..n), n=0..14) ; [From R. J. Mathar, Nov 03 2009]
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CROSSREFS
| Cf. A138034, A135936, A137276 (row-reversed).
Sequence in context: A157424 A144961 A144627 * A080733 A080732 A088568
Adjacent sequences: A135926 A135927 A135928 * A135930 A135931 A135932
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KEYWORD
| sign,tabl
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mar 09 2008
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EXTENSIONS
| Extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 03 2009
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