A192174 -id:A192174 - cp's OEIS frontend

A112467 Riordan array ((1-2x)/(1-x), x/(1-x)).

1, -1, 1, -1, 0, 1, -1, -1, 1, 1, -1, -2, 0, 2, 1, -1, -3, -2, 2, 3, 1, -1, -4, -5, 0, 5, 4, 1, -1, -5, -9, -5, 5, 9, 5, 1, -1, -6, -14, -14, 0, 14, 14, 6, 1, -1, -7, -20, -28, -14, 14, 28, 20, 7, 1, -1, -8, -27, -48, -42, 0, 42, 48, 27, 8, 1, -1, -9, -35, -75, -90, -42, 42, 90, 75, 35, 9, 1, -1, -10, -44, -110, -165, -132, 0, 132, 165, 110

Offset: 0

Paul Barry, Sep 06 2005

Row sums are A000007. Diagonal sums are -F(n-2). Inverse is A112468. T(2n,n)=0.
(-1,1)-Pascal triangle. - Philippe Deléham, Aug 07 2006
Apart from initial term, same as A008482. - Philippe Deléham, Nov 07 2006
Each column equals the cumulative sum of the previous column. - Mats Granvik, Mar 15 2010
Reading along antidiagonals generates in essence rows of A192174. - Paul Curtz, Oct 02 2011
Triangle T(n,k), read by rows, given by (-1,2,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 01 2011

			Triangle starts:
    1;
   -1,  1;
   -1,  0,   1;
   -1, -1,   1,   1;
   -1, -2,   0,   2,   1;
   -1, -3,  -2,   2,   3,   1;
   -1, -4,  -5,   0,   5,   4,  1;
   -1, -5,  -9,  -5,   5,   9,  5,  1;
   -1, -6, -14, -14,   0,  14, 14,  6,  1;
   -1, -7, -20, -28, -14,  14, 28, 20,  7,  1;
   -1, -8, -27, -48, -42,   0, 42, 48, 27,  8, 1;
   -1, -9, -35, -75, -90, -42, 42, 90, 75, 35, 9, 1;
  ...
From _Paul Barry_, Apr 08 2011: (Start)
Production matrix begins:
   1,  1,
  -2, -1,  1,
   2,  0, -1,  1,
  -2,  0,  0, -1,  1,
   2,  0,  0,  0, -1,  1,
  -2,  0,  0,  0,  0, -1,  1,
   2,  0,  0,  0,  0,  0, -1,  1
  ... (End)

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Elena Barcucci, Antonio Bernini, Stefano Bilotta, and Renzo Pinzani, Restricting Dyck Paths and 312-avoiding Permutations, arXiv:2307.02837 [math.CO], 2023. Mentions this sequence.
Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
Emeric Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.
D. Foata and G.-N. Han, The doubloon polynomial triangle, Ram. J. 23 (2010), 107-126.
Jack Ramsay, On Arithmetical Triangles, The Pulse of Long Island, June 1965. [Mentions application to design of antenna arrays. Annotated scan.]

Same first 3 rows as in A054525.

Cf. A008482, A037012, A080232, A112466, A112467, A174293, A174294, A174295, A174296, A174297.

[n eq 0 select 1 else (2*k-n)*Binomial(n,k)/n: k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 04 2019

seq(seq( `if`(n=0, 1, (2*k-n)*binomial(n,k)/n), k=0..n), n=0..10); # G. C. Greubel, Dec 04 2019

T[n_, k_]= If[n==0, 1, ((2*k-n)/n)*Binomial[n, k]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Roger L. Bagula, Feb 16 2009; modified by G. C. Greubel, Dec 04 2019 *)

T(n, k) = if(n==0, 1, (2*k-n)*binomial(n,k)/n ); \\ G. C. Greubel, Dec 04 2019

def T(n, k):
    if (n==0): return 1
    else: return (2*k-n)*binomial(n,k)/n
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 04 2019

Number triangle T(n, k) = binomial(n, n-k) - 2*binomial(n-1, n-k-1).
Sum_{k=0..n} T(n, k)*x^k = (x-1)*(x+1)^(n-1). - Philippe Deléham, Oct 03 2005
T(n,k) = ((2*k-n)/n)*binomial(n, k), with T(0,0)=1. - Roger L. Bagula, Feb 16 2009; modified by G. C. Greubel, Dec 04 2019
T(n,k) = T(n-1,k-1) + T(n-1,k) with T(0,0)=1, T(1,0)=-1, T(n,k)=0 for k>n or for n<0. - Philippe Deléham, Nov 01 2011
G.f.: (1-2x)/(1-(1+y)*x). - Philippe Deléham, Dec 15 2011
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A133494(n), A081294(n), A005053(n), A067411(n), A199661(n), A083233(n) for x = 1, 2, 3, 4, 5, 6, 7, respectively. - Philippe Deléham, Dec 15 2011
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(-1 - x + x^2/2! + x^3/3!) = -1 - 2*x - 2*x^2/2! + 5*x^4/4! + 14*x^5/5! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 21 2014
Sum_{k=0..n} T(n,k) = 0^n = A000007(n). - G. C. Greubel, Dec 04 2019

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

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.

Original entry on oeis.org

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.

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A112467 Riordan array ((1-2x)/(1-x), x/(1-x)).

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

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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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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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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).