A194084 -id:A194084 - cp's OEIS frontend

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.

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

Offset: 0

N. J. A. Sloane, Mar 09 2008

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

			The coefficients and 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

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.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, and Minghao Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.

Cf. A049310, A053119, A138034, A135936, A137276 (row-reversed), A192011, A194084, A219795.

A053119:= func< n,k | (1/2)*(-1)^Floor(3*k/2)*(1+(-1)^k)*Binomial(n - Floor(k/2), n-k) >;
A135929:= func< n,k | n eq 0 select 1 else A053119(n, k) + 3*A053119(n-2, k-2) >;
[A135929(n,k): k in [0..n], n in [0..16]]; // G. C. Greubel, Apr 24 2023

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

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 *)
(* Second program *)
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 *)

def A053119(n,k): return (-1)^(3*k/2)*((k+1)%2)*binomial(n-k/2, n-k)
def A135929(n,k): return 1 if (n==0) else A053119(n, k) + 3*A053119(n-2, k-2)
flatten([[A135929(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Apr 24 2023

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
From G. C. Greubel, Apr 24 2023: (Start)
T(n, k) = A053119(n, k) + 3*A053119(n-2, k-2), with T(0,0) = 1.
Sum_{k=0..n} T(n, k) = A138034(n). (End)

Extended by R. J. Mathar, Nov 03 2009

A192174 Triangle T(n,k) of the coefficients [x^(n-k)] of the polynomial p(0,x)=-1, p(1,x)=x and p(n,x) = x*p(n-1,x) - p(n-2,x) in row n, column k.

Original entry on oeis.org

-1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, -1, 0, -1, 1, 0, -2, 0, -1, 0, 1, 0, -3, 0, 0, 0, 1, 1, 0, -4, 0, 2, 0, 2, 0, 1, 0, -5, 0, 5, 0, 2, 0, -1, 1, 0, -6, 0, 9, 0, 0, 0, -3, 0, 1, 0, -7, 0, 14, 0, -5, 0, -5, 0, 1, 1, 0, -8, 0, 20, 0, -14, 0, -5, 0, 4, 0

Offset: 0

Paul Curtz, Jun 24 2011

Consider the Catalan triangle A009766 antisymmetrically extended by a mirror along the diagonal (see also A176239):
0, -1, -1, -1, -1,  -1,  -1,   -1,
1,  0, -1, -2, -3,  -4,  -5,   -6,
1,  1,  0, -2, -5,  -9, -14,  -20,
1,  2,  2,  0, -5, -14, -28,  -48,
1,  3,  5,  5,  0, -14, -42,  -90,
1,  4,  9, 14, 14,   0, -42, -132,
1,  5, 14, 28, 42,  42,   0, -132,
1,  6, 20, 48, 90, 132, 132,    0.
The rows in this array are essentially the columns of T(n,k).

			Triangle begins
  -1;      # -1
   1,  0;      # x
   1,  0,  1;      # x^2+1
   1,  0,  0,  0;      # x^3
   1,  0, -1,  0, -1;      # x^4-x^2-1
   1,  0, -2,  0, -1,  0;
   1,  0, -3,  0,  0,  0,  1;
   1,  0, -4,  0,  2,  0,  2,  0;
   1,  0, -5,  0,  5,  0,  2,  0, -1;
   1,  0, -6,  0,  9,  0,  0,  0, -3,  0;
   1,  0, -7,  0, 14,  0, -5,  0, -5,  0,  1;
   1,  0, -8,  0, 20,  0,-14,  0, -5,  0,  4,  0;
   1,  0, -9,  0, 27,  0,-28,  0,  0,  0,  9,  0, -1;

Cf. A023443, A080956 and A000096, A129936 and A005586, A005587.

Cf. A194084. - Paul Curtz, Aug 16 2011

p:= proc(n,x) option remember: if n=0 then -1 elif n=1 then x elif n>=2 then x*procname(n-1,x)-procname(n-2,x) fi: end: A192174 := proc(n,k): coeff(p(n,x),x,n-k): end: seq(seq(A192174(n,k),k=0..n), n=0..11); # Johannes W. Meijer, Aug 21 2011

Sum_{k=0..n} T(n,k) = A057079(n-1).

Apparently T(3s,2s-2) = (-1)^(s+1)*A000245(s), s >= 1.

cp's OEIS Frontend

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.

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