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A135929 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. 20
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; text; internal format)
OFFSET

0,6

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

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.

LINKS

Table of n, a(n) for n=0..91.

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]

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

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) ; # R. J. Mathar, Nov 03 2009

MATHEMATICA

p[0, _] = 1; p[1, x_] := x; p[2, x_] := x^2+2; p[n_, x_] := p[n, x] = x*p[n-1, x] - p[n-2, x]; row[n_] := CoefficientList[p[n, x], x]; Table[row[n] // Reverse, {n, 0, 13}] // Flatten (* Jean-François Alcover, Nov 26 2012, after Paul Curtz's formula *)

(* Or : *) p=1; q=2; t[_, 0]=p; t[2, 2]=q; t[_, _?OddQ]=0; t[n_, k_] /; k > n = 0; t[n_ /; n >= 0, k_ /; k >= 0] := t[n, k] = t[n-1, k] - t[n-2, k-2]; Table[t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 27 2012, from recurrence *)

CROSSREFS

Cf. A138034, A135936, A137276 (row-reversed), A194084, A219795.

Sequence in context: A157424 A144961 A144627 * A080733 A080732 A215036

Adjacent sequences:  A135926 A135927 A135928 * A135930 A135931 A135932

KEYWORD

sign,tabl

AUTHOR

N. J. A. Sloane, Mar 09 2008

EXTENSIONS

Extended by R. J. Mathar, Nov 03 2009

STATUS

approved

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Last modified November 19 11:04 EST 2017. Contains 294936 sequences.