A192011 -id:A192011 - cp's OEIS frontend

A194084 Triangle read by rows: a(n)=A135929(n) + A192011(n). Row n gives coefficients of polynomials BC(n,x) in order of decreasing exponents.

0, 3, 0, 3, 0, 3, 3, 0, 0, 0, 3, 0, -3, 0, -3, 3, 0, -6, 0, -3, 0, 3, 0, -9, 0, 0, 0, 3, 3, 0, -12, 0, 6, 0, 6, 0, 3, 0, -15, 0, 15, 0, 6, 0, -3, 3, 0, -18, 0, 27, 0, 0, 0, -9, 0

Offset: 0

Paul Curtz, Aug 14 2011

0,
3, 0,
3, 0,  3,
3, 0,  0, 0,
3, 0, -3, 0, -3,
3, 0, -6, 0, -3, 0.
Multiples of 3.
Row sum (from the second) is period 6: 3*A057079(n),"from" A057083 (scaled Chebyshev U(n,x)).
If a(0)=-3, a(n)=3*A192174(n).

			a(0)=1-1=0, a(1)=1+2=3, a(2)=0+0=0.

BC(0,x)=0, BC(1,x)=3*x, BC(2,x)=3*x^2+3, BC(n,x)=x*BC(n-1,x) - BC(n-2,x), n > 2.

A198862 Sum of the n-th antidiagonal in the triangle A192011.

Original entry on oeis.org

-1, 2, 2, 2, 3, 1, -1, -3, -6, -7, -6, -3, 3, 10, 16, 19, 16, 6, -10, -29, -45, -51, -41, -12, 33, 84, 125, 137, 104, 20, -105, -242, -346, -366, -261, -19, 327, 693, 954, 973, 646, -47, -1001, -1974, -2620, -2573, -1572, 402, 3022

Offset: 0

Paul Curtz, Oct 30 2011

The current sequence and its successive differences are:
-1, 2,  2,  2,  3,  1, -1 ,-3, -6, -7, ...
3, 0, 0, 1, -2, -2, -2, -3, -1, 1, 3, 6, 7, 6, 3, -3, -10, -16, ...
-3, 0, 1, -3, 0, 0, -1, 2, 2, 2, 3, 1, -1, -3, -6, -7, -6, ...
3, 1, -4, 3, 0, -1, 3, 0, 0, 1, -2, -2, -2, -3, -1, 1, 3, 6, 7, ...
-2, -5, 7, -3, -1, 4, -3, 0, 1, -3, 0, 0, -1, 2, 2, 2, 3, 1, ...
-3, 12, -10, 2, 5, -7, 3, 1, -4, 3, 0, -1, 3, 0, 0, 1, -2, ...
15, -22, 12, 3, -12, 10, -2, -5, 7, -3, -1, 4, -3, 0, ...
-37, 34, -9, -15, 22, -12, -3, 12, -10, 2, 5, -7, 3, 1, -4, ...
Each row obeys the same linear recurrence and is a version of the row 4 lines farther up in the same array shifted right by 12 places.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,-1).

A198862 := proc(n)
        add( A192011(n-k,k),k=0..floor(n/2)) ;
end proc:
seq(A198862(n),n=0..80) ; # R. J. Mathar, Nov 03 2011

a(n) = Sum_{k=0..floor(n/2)} A192011(n-k,k).
a(n) = a(n-1) - a(n-4), n > 3.
From R. J. Mathar, Nov 02 2011: (Start)
G.f.: (-1 + 3*x) / (1 - x + x^4).
a(n) = 3*A099530(n-1) - A099530(n). (End)

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.

Original entry on oeis.org

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

A053119 Triangle of coefficients of Chebyshev's S(n,x) polynomials (exponents in decreasing order).

Original entry on oeis.org

1, 1, 0, 1, 0, -1, 1, 0, -2, 0, 1, 0, -3, 0, 1, 1, 0, -4, 0, 3, 0, 1, 0, -5, 0, 6, 0, -1, 1, 0, -6, 0, 10, 0, -4, 0, 1, 0, -7, 0, 15, 0, -10, 0, 1, 1, 0, -8, 0, 21, 0, -20, 0, 5, 0, 1, 0, -9, 0, 28, 0, -35, 0, 15, 0, -1, 1, 0, -10, 0, 36, 0, -56, 0, 35, 0, -6, 0, 1, 0, -11, 0, 45, 0, -84, 0, 70, 0, -21, 0, 1

Offset: 0

Wolfdieter Lang

These polynomials also give the determinant of the tridiagonal matrix having x on the diagonal and -1 next to these x. - M. F. Hasler, Oct 15 2019

The polynomial S(n,x) is the character of the irreducible (n+1) dimensional representation of the Lie algebra sl_2 when x is the character of irreducible 2-dimesional representation. - Leonid Bedratyuk, Oct 28 2023

			The triangle begins:
n\m 0  1   2  3   4  5   6  7   8  9  10 ...
0:  1
1:  1  0
2:  1  0  -1
3:  1  0  -2  0
4:  1  0  -3  0   1
5:  1  0  -4  0   3  0
6:  1  0  -5  0   6  0  -1
7:  1  0  -6  0  10  0  -4  0
8:  1  0  -7  0  15  0 -10  0   1
9:  1  0  -8  0  21  0 -20  0   5  0
10: 1  0  -9  0  28  0 -35  0  15  0  -1
... Reformatted. - _Wolfdieter Lang_, Dec 17 2013
E.g., fourth row (n=3) corresponds to polynomial S(3,x)= x^3-2*x.
Triangle of absolute values of coefficients (coefficients of Fibonacci polynomials) with exponents in increasing order begins:
[1]
[0, 1]
[1, 0, 1]
[0, 2, 0, 1]
[1, 0, 3, 0, 1]
[0, 3, 0, 4, 0, 1]
[1, 0, 6, 0, 5, 0, 1]
[0, 4, 0, 10, 0, 6, 0, 1]
[1, 0, 10, 0, 15, 0, 7, 0, 1]
[0, 5, 0, 20, 0, 21, 0, 8, 0, 1]
See A162515 for the Fibonacci polynomials with reversed row entries, starting there with row 1. - _Wolfdieter Lang_, Dec 16 2013

D. S. Mitrinovic, Analytic Inequalities, Springer-Verlag, 1970; p. 232, Sect. 3.3.38.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

T. D. Noe, Rows n=0..100 of triangle, flattened
Tom Copeland, Addendum to Elliptic Lie Triad
Index entries for sequences related to Chebyshev polynomials.

Row sums give A000045. Reflection of A049310.

Cf. A162515. - Wolfdieter Lang, Dec 16 2013

A053119 := (n, k) -> if k::even then (-1)^binomial(k, 2)*binomial(n - k/2, k/2)
else 0 fi: seq(seq(A053119(n, k), k = 0..n), n = 0..11); # Peter Luschny, Jul 20 2024

ChebyshevS[n_, x_] := ChebyshevU[n, x/2]; Flatten[ Table[ Reverse[ CoefficientList[ ChebyshevS[n, x], x]], {n, 0, 12}]] (* Jean-François Alcover, Nov 25 2011 *)

tabl(nn) = for (n=0, nn, print(Vec(polchebyshev(n, 2, x/2)))); \\ Michel Marcus, Jan 14 2016

a(n,m) = A049310(n,n-m).
G.f. for row polynomials S(n,x) (signed triangle): 1/(1-x*z+z^2).
Unsigned triangle |a(n,m)| has Fibonacci polynomials F(n+1,x) as row polynomials with G.f. 1/(1-x*z-z^2).
a(n, m) := 0 if n < m or m odd, else ((-1)^(3*m/2))*binomial(n-m/2, n-m); a(n, m) = a(n-1, m) - a(n-2, m-2), a(n, -2) := 0 =: a(n, -1), a(0, 0) = 1, a(n, m) = 0 if n < m or m odd.
G.f. for m-th column (signed triangle): (-1)^(3*m/2)*x^m/(1-x)^(m/2+1) if m >= 0 is even else 0.
Recurrence for the (unsigned) Fibonacci polynomials: F[1]=1, F[2]=x; for n>2, F[n] = x*F[n-1]+F[n-2].
a = 2*A192011 - 3*A192174. - Thomas Baruchel, Jun 02 2018
Recurrence for the polynomials S(n) = x S(n-1) - S(n-2); S(0) = 1, S(1) = x. - M. F. Hasler, Oct 15 2019

A219795 Sum of the absolute values of the antidiagonals of the triangle A135929(n) companion. See the comment.

Original entry on oeis.org

2, 2, 2, 2, 3, 3, 5, 7, 10, 11, 16, 23, 33, 44, 58, 81, 114, 158, 212, 293, 407, 565, 777, 1064, 1471, 2036, 2813, 3863, 5334, 7370, 10183, 14046, 19356, 26726, 36909, 50955, 70251, 96977, 133886, 184841, 255092

Offset: 0

Paul Curtz, Nov 28 2012

nonn

The companion to A135929(n) is the triangle
2;
2, 0;
2, 0,  1;
2, 0, -1, 0;
2, 0, -3, 0, -1;
2, 0, -5, 0,  0, 0;
2, 0, -7, 0,  3, 0, 1;
2, 0, -9, 0,  8, 0, 1, 0;
(A192011(n) beginning with 2 instead of -1).
Consider a(1),a(5),a(10),a(14), that is, a(A193910(n) -1).
a(1)+a(4)-a(5) = 2, a(5)+a(8)-a(9) = 2, a(10)+a(13)-a(14) = 2, a(14)+a(17)-a(18) = 4, a(19)+a(22)-a(23) = 6, a(23)+a(26)-a(27) = 14, yields 2,2,2,4,6,14,24,60,... = 2*A047749(n) or 2, followed with A116637(n+1).

			a(0)=2, a(1)=2, a(2)=2+0, a(3)=2+0, a(4)=2+0+1, a(5)=2+0+1.

A219795 := proc(n)
    if n=0 then
        2;
    else
        add(abs(A192011(n-k,k)),k=0..floor(n/2)) ;
    end if;
end proc: # R. J. Mathar, Jan 06 2013

a(n) = sum abs ( [k=0..floor(n/2)] A192011(n-k,k) ), a(0)=2.

a(24)-a(40) from Jean-Francois Alcover, Nov 28 2012

A174559 Triangle T(n,k)of the coefficients [x^(n-k)] of the polynomials q(0,x)=-1, q(1,x)=3x, q(n,x)=xq(n-1,x)-q(n-2,x) in row n,column k. A companion to A193002(n).

Original entry on oeis.org

-1, 3, 0, 3, 0, 1, 3, 0, -2, 0, 3, 0, -5, 0, -1, 3, 0, -8, 0, 1, 0, 3, 0, -11, 0, 6, 0, 1, 3, 0, -14, 0, 14, 0, 0, 0, 3, 0, -17, 0, 25, 0, -6, 0, -1, 3, 0, -20, 0, 39, 0, -20, 0, -1, 0, 3, 0, -23, 0, 56, 0, -45, 0, 5, 0, 1

Offset: 0

Paul Curtz, Aug 20 2011

a(n)=
-1,   :-1,
3,  0,   :3*x,
3,  0,   1,   :3*x^2+1,
3,  0,  -2,  0,   :3*x^3-2*x,
3,  0,  -5,  0, -1,
3,  0,  -8,  0,  1, 0
3,  0,  -11, 0,  6, 0, 1.
Row sum=period 6:repeat -1, 3, 4, 1, -3, 4=-A117378(n)=A117378(n+3).

Cf. A192011.

a(n) + A193002(n)=4*A192174(n).

a(n) - A193002(n)=2*A053119(n), Chebyshev's S(n,x).

A195662 Triangle T(n,k) read by rows: T(0,0)= -3, T(1,0)= 2, T(1,1) = 0 and T(n,k) = T(n-1,k) -T(n-2,k-2) otherwise.

Original entry on oeis.org

-3, 2, 0, 2, 0, 3, 2, 0, 1, 0, 2, 0, -1, 0, -3, 2, 0, -3, 0, -4, 0, 2, 0, -5, 0, -3, 0, 3, 2, 0, -7, 0, 0, 0, 7, 0, 2, 0, -9, 0, 5, 0, 10, 0, -3, 2, 0, -11, 0, 12, 0, 10, 0, -10, 0, 2, 0, -13, 0, 21, 0, 5, 0, -20, 0, 3, 2, 0, -15, 0, 32, 0, -7, 0, -30, 0, 13, 0

Offset: 0

Paul Curtz, Sep 22 2011

In the notation of A195673, this defines polynomials P(n,x,p=-3,q=2), where p and q are the values of the constant and linear order for n=0 and 1.
Row sums -- the value P(n,1,-3,2) of the polynomial -- are A130848(n+5).
For general seed values in the two top rows of the triangle, the recurrence T(n,k) = T(n-1,k) - T(n-2,k-2) defines the triangle
p;
q,   0;
q,   0,      -p;
q,   0,    -p-q, 0;
q,   0,   -p-2q, 0,     p;
q,   0,   -p-3q, 0,  2p+q,  0;
and a companion triangle by adding 1 to both seed values:
p+1;
q+1, 0;
q+1, 0,    -p-1;
q+1, 0,  -p-q-2, 0;
q+1, 0, -p-2q-3, 0,    p+1;
q+1, 0, -p-3q-4, 0, 2p+q+3, 0;
The point-by-point difference between two companions is P(n,x,p+1,q+1) - P(n,x,p,q) = S(n,x) as represented (with increasing exponents) by A053119.
Examples of such triangles are A053119 (p=q=1), A192575 (p=1, q=2),
A162514 (p=2, q=1, up to a sign factor), A192011 (p=-1, q=2), A135929 (p=-2, q=1, apart from a irregular leading T(0,0)).

			The first few rows are
-3;
2, 0;
2, 0,   3;
2, 0,   1, 0;
2, 0,  -1, 0, -3;
2, 0,  -3, 0, -4, 0;
2, 0,  -5, 0, -3, 0,  3;
2, 0,  -7, 0,  0, 0,  7, 0;
2, 0,  -9, 0,  5, 0, 10, 0,  -3;
2, 0, -11, 0, 12, 0, 10, 0, -10, 0;
2, 0, -13, 0, 21, 0,  5, 0, -20, 0, 3;

Cf. A135929, A192011.

p = -3; q = 2; t[0, 0] = p; t[, 0] = 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, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 27 2012 *)

T(n,0) = 2 (n>0).
T(n,2) = -A060747(n-3), n>2.
T(n,4) = A028347(n-5), n>6.
T(2n,2n) = -3*(-1)^n ; T(n, 2k-1) = 0 ; T(2n+1,2n) = -(3n-2)*(-1)^n.  - M. F. Hasler, Sep 28 2011

A195673 Triangle T(n,k) read by rows: T(0,0)=-2, T(1,0)=3, T(1,1)=0 and T(n,k) = T(n-1,k)-T(n-2,k-2) otherwise.

Original entry on oeis.org

-2, 3, 0, 3, 0, 2, 3, 0, -1, 0, 3, 0, -4, 0, -2, 3, 0, -7, 0, -1, 0, 3, 0, -10, 0, 3, 0, 2, 3, 0, -13, 0, 10, 0, 3, 0, 3, 0, -16, 0, 20, 0, 0, 0, -2, 3, 0, -19, 0, 33, 0, -10, 0, -5, 0, 3, 0, -22, 0, 49, 0, -30, 0, -5, 0, 2, 3, 0, -25, 0, 68

Offset: 0

Paul Curtz, Sep 23 2011

Obviously T(n,k) = 0 for all odd k.

Conjecture: The polynomials p(n,x) = sum_{k=0..n} T(n,k)*x^(n-k) based on this simple recurrence for other initial constant values of T(0,0)=p and T(1,0)=q are related to the S-polynomials of A053119: p(n,x,p+1,q+1)-p(n,x,p,q) = S(n,x).

			-2;
3, 0;
3, 0,   2;
3, 0,  -1, 0;
3, 0,  -4, 0, -2;
3, 0,  -7, 0, -1, 0;
3, 0, -10, 0,  3, 0, 2;
3, 0, -13, 0, 10, 0, 3, 0.

Cf. A195662, A192011 (p=-1, q=2), A135929 (p=-2, q=1).

cp's OEIS Frontend

A194084 Triangle read by rows: a(n)=A135929(n) + A192011(n). Row n gives coefficients of polynomials BC(n,x) in order of decreasing exponents.

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A198862 Sum of the n-th antidiagonal in the triangle A192011.

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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.

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A053119 Triangle of coefficients of Chebyshev's S(n,x) polynomials (exponents in decreasing order).

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A219795 Sum of the absolute values of the antidiagonals of the triangle A135929(n) companion. See the comment.

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A174559 Triangle T(n,k)of the coefficients [x^(n-k)] of the polynomials q(0,x)=-1, q(1,x)=3*x, q(n,x)=x*q(n-1,x)-q(n-2,x) in row n,column k. A companion to A193002(n).

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A195662 Triangle T(n,k) read by rows: T(0,0)= -3, T(1,0)= 2, T(1,1) = 0 and T(n,k) = T(n-1,k) -T(n-2,k-2) otherwise.

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A195673 Triangle T(n,k) read by rows: T(0,0)=-2, T(1,0)=3, T(1,1)=0 and T(n,k) = T(n-1,k)-T(n-2,k-2) otherwise.

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A174559 Triangle T(n,k)of the coefficients [x^(n-k)] of the polynomials q(0,x)=-1, q(1,x)=3x, q(n,x)=xq(n-1,x)-q(n-2,x) in row n,column k. A companion to A193002(n).