A157751 -id:A157751 - cp's OEIS frontend

A192313 Constant term of the reduction of n-th polynomial at A157751 by x^2->x+1.

1, 2, 5, 13, 34, 91, 247, 680, 1893, 5319, 15056, 42867, 122605, 351898, 1012729, 2920521, 8435362, 24392655, 70599403, 204472264

Offset: 1

Clark Kimberling, Jun 28 2011

nonn

For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

			The first five polynomials at A157751 and their reductions are as follows:
p0(x)=1 -> 1
p1(x)=2+x -> 2+x
p2(x)=4+2x+x^2 -> 5+3x
p3(x)=8+4x+4x^2+x^3 -> 13+10x
p4(x)=16+8x+12x^2+4x^3+x^4 -> 34+31x.
From these, we read
A192313=(1,2,5,13,34,...) and A192314=(0,1,3,19,31,...)

Cf. A192232, A192337.

q[x_] := x + 1;
p[0, x_] := 1;
p[n_, x_] := (x + 1)*p[n - 1, x] + p[n - 1, -x] /; n > 0  (* A157751 *)
Table[Simplify[p[n, x]], {n, 0, 5}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 20}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 20}]
  (* A192313 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 20}]
  (* A192337 *)

Empirical G.f.: x*(x+1)*(x^2-3*x+1)/(x^4+6*x^3+x^2-4*x+1). [Colin Barker, Nov 13 2012]

A192338 Constant term of the reduction of n-th polynomial at A157751 by x^2->x+2.

Original entry on oeis.org

1, 2, 6, 18, 54, 166, 514, 1610, 5078, 16118, 51394, 164474, 527798, 1697254, 5466498, 17627370, 56892246, 183742358

Offset: 1

Clark Kimberling, Jun 28 2011

nonn

For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232.

			The first five polynomials at A157751 and their reductions are as follows:
p0(x)=1 -> 1
p1(x)=2+x -> 2+x
p2(x)=4+2x+x^2 -> 6+3x
p3(x)=8+4x+4x^2+x^3 -> 18+11x
p4(x)=16+8x+12x^2+4x^3+x^4 -> 54+37x.
From these, we read
A192338=(1,2,6,18,54,...) and A192339=(0,1,3,11,37,...)

Cf. A192232, A192339.

q[x_] := x + 2;
p[0, x_] := 1;
p[n_, x_] := (x + 1)*p[n - 1, x] + p[n - 1, -x] /;
  n > 0  (* polynomials defined at A157751 *)
Table[Simplify[p[n, x]], {n, 0, 5}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
   x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0,16}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 16}]
(* A192338 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 16}]
(* A192339 *)

Conjecture: a(n) = 4*a(n-1)+a(n-2)-10*a(n-3)-4*a(n-4). G.f.: x*(2*x^3-3*x^2-2*x+1) / ((x^2+2*x-1)*(4*x^2+2*x-1)). [Colin Barker, Nov 22 2012]

A192337 Coefficient of x in the reduction of n-th polynomial at A157751 by x^2->x+1.

Original entry on oeis.org

0, 1, 3, 10, 31, 95, 286, 853, 2525, 7436, 21815, 63821, 186328, 543165, 1581591, 4601410, 13378731, 38880803, 112954430, 328063121

Offset: 1

Clark Kimberling, Jun 28 2011

nonn

For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

Cf. A192232, A192313.

Mathematica
```
(See A192313.)
```

Empirical G.f.: -x^2*(x^2+x-1)/(x^4+6*x^3+x^2-4*x+1). [Colin Barker, Nov 13 2012]

A192339 Coefficient of x in the reduction of n-th polynomial at A157751 by x^2->x+2.

Original entry on oeis.org

0, 1, 3, 11, 37, 125, 415, 1371, 4501, 14725, 48031, 156355, 508197, 1649933, 5352255, 17351563, 56226389, 182134837

Offset: 1

Clark Kimberling, Jun 28 2011

nonn

For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232.

Cf. , A192232, A192313, A192338.

Mathematica
```
(See A192338.)
```

Empirical G.f.: -x^2*(x+1)*(2*x-1)/((x^2+2*x-1)*(4*x^2+2*x-1)). [Colin Barker, Nov 13 2012]

A130777 Coefficients of first difference of Chebyshev S polynomials.

Original entry on oeis.org

1, -1, 1, -1, -1, 1, 1, -2, -1, 1, 1, 2, -3, -1, 1, -1, 3, 3, -4, -1, 1, -1, -3, 6, 4, -5, -1, 1, 1, -4, -6, 10, 5, -6, -1, 1, 1, 4, -10, -10, 15, 6, -7, -1, 1, -1, 5, 10, -20, -15, 21, 7, -8, -1, 1, -1, -5, 15, 20, -35, -21, 28, 8, -9, -1, 1, 1, -6, -15, 35, 35, -56, -28, 36, 9, -10, -1, 1

Offset: 0

Philippe Deléham, Jul 14 2007

Inverse of triangle in A061554.
Signed version of A046854.
From Paul Barry, May 21 2009: (Start)
Riordan array ((1-x)/(1+x^2),x/(1+x^2)).
This triangle is the coefficient triangle for the Hankel transforms of the family of generalized Catalan numbers that satisfy a(n;r)=r*a(n-1;r)+sum{k=1..n-2, a(k)*a(n-1-k;r)}, a(0;r)=a(1;r)=1. The Hankel transform of a(n;r) is h(n)=sum{k=0..n, T(n,k)*r^k} with g.f. (1-x)/(1-r*x+x^2). These sequences include A086246, A000108, A002212. (End)
From Wolfdieter Lang, Jun 11 2011: (Start)
The Riordan array ((1+x)/(1+x^2),x/(1+x^2)) with entries Phat(n,k)= ((-1)^(n-k))*T(n,k) and o.g.f. Phat(x,z)=(1+z)/(1-x*z+z^2) for the row polynomials Phat(n,x) is related to Chebyshev C and S polynomials as follows.
Phat(n,x) = (R(n+1,x)-R(n,x))/(x+2) = S(2*n,sqrt(2+x))
  with R(n,x)=C_n(x) in the Abramowitz and Stegun notation, p. 778, 22.5.11. See A049310 for the S polynomials. Proof from the o.g.f.s.
Recurrence for the row polynomials Phat(n,x):
  Phat(n,x) = x*Phat(n-1,x) - Phat(n-2,x) for n>=1; Phat(-1,x)=-1, Phat(0,x)=1.
The A-sequence for this Riordan array Phat (see the W. Lang link under A006232 for A- and Z-sequences for Riordan matrices) is given by 1, 0, -1, 0, -1, 0, -2, 0, -5,.., starting with 1 and interlacing the negated A000108 with zeros (o.g.f. 1/c(x^2) = 1-c(x^2)*x^2, with the o.g.f. c(x) of A000108).
The Z-sequence has o.g.f. sqrt((1-2*x)/(1+2*x)), and it is given by A063886(n)*(-1)^n.
The A-sequence of the Riordan array T(n,k) is identical with the one for the Riordan array Phat, and the Z-sequence is -A063886(n).
(End)
The row polynomials P(n,x) are the characteristic polynomials of the adjacency matrices of the graphs which look like P_n (n vertices (nodes), n-1 lines (edges)), but vertex no. 1 has a loop. - Wolfdieter Lang, Nov 17 2011
From Wolfdieter Lang, Dec 14 2013: (Start)
The zeros of P(n,x) are x(n,j) = -2*cos(2*Pi*j/(2*n+1)), j=1..n. From P(n,x) = (-1)^n*S(2*n,sqrt(2-x)) (see, e.g., the Lemma 6 of the W. Lang link).
The discriminants of the P-polynomials are given in A052750. (End)

			The triangle T(n,k) begins:
n\k  0   1   1   3    4    5    6    7    8    9  10  11  12  13 14 15 ...
0:   1
1:  -1   1
2:  -1  -1   1
3:   1  -2  -1   1
4:   1   2  -3  -1    1
5:  -1   3   3  -4   -1    1
6:  -1  -3   6   4   -5   -1    1
7:   1  -4  -6  10    5   -6   -1    1
8:   1   4 -10 -10   15    6   -7   -1    1
9:  -1   5  10 -20  -15   21    7   -8   -1    1
10: -1  -5  15  20  -35  -21   28    8   -9   -1   1
11:  1  -6 -15  35   35  -56  -28   36    9  -10  -1   1
12:  1   6 -21 -35   70   56  -84  -36   45   10 -11  -1   1
13: -1   7  21 -56  -70  126   84 -120  -45   55  11 -12  -1   1
14: -1  -7  28  56 -126 -126  210  120 -165  -55  66  12 -13  -1  1
15:  1  -8 -28  84  126 -252 -210  330  165 -220 -66  78  13 -14 -1  1
...  reformatted and extended - _Wolfdieter Lang_, Jul 31 2014.
---------------------------------------------------------------------------
From _Paul Barry_, May 21 2009: (Start)
Production matrix is
-1, 1,
-2, 0, 1,
-2, -1, 0, 1,
-4, 0, -1, 0, 1,
-6, -1, 0, -1, 0, 1,
-12, 0, -1, 0, -1, 0, 1,
-20, -2, 0, -1, 0, -1, 0, 1,
-40, 0, -2, 0, -1, 0, -1, 0, 1,
-70, -5, 0, -2, 0, -1, 0, -1, 0, 1 (End)
Row polynomials as first difference of S polynomials:
P(3,x) = S(3,x) - S(2,x) = (x^3 - 2*x) - (x^2 -1) = 1 - 2*x - x^2 +x^3.
Alternative triangle recurrence (see a comment above): T(6,2) = T(5,2) + T(5,1) = 3 + 3 = 6. T(6,3) = -T(5,3) + 0*T(5,1) = -(-4) = 4. - _Wolfdieter Lang_, Jul 31 2014

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964. Tenth printing, Wiley, 2002 (also electronically available).

Hyeong-Kwan Ju, On the sequence generated by a certain type of matrices, Honam Math. J. 39, No. 4, 665-675 (2017), Theorem 2.16.
Wolfdieter Lang, The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon, arXiv:1210.1018 [math.GR], 2012-2017; see Definition 1, Lemma 6 and Remark 4.
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), p. 22-31 (formula 5).
Index entries for sequences related to Chebyshev polynomials.

Cf. A066170, A046854, A057077 (first column).

Row sums: A010892(n+1); repeat(1,0,-1,-1,0,1). Alternating row sums: A061347(n+2); repeat(1,-2,1).

A130777 := proc(n,k): (-1)^binomial(n-k+1,2)*binomial(floor((n+k)/2),k) end: seq(seq(A130777(n,k), k=0..n), n=0..11); # Johannes W. Meijer, Aug 08 2011

T[n_, k_] := (-1)^Binomial[n - k + 1, 2]*Binomial[Floor[(n + k)/2], k];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 14 2017, from Maple *)

@CachedFunction
def A130777(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = A130777(n-1,k) if n==1 else 0
    return A130777(n-1,k-1) - A130777(n-2,k) - h
for n in (0..9): [A130777(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

Number triangle T(n,k) = (-1)^C(n-k+1,2)*C(floor((n+k)/2),k). - Paul Barry, May 21 2009
From Wolfdieter Lang, Jun 11 2011: (Start)
Row polynomials: P(n,x) = sum(k=0..n, T(n,k)*x^k) = R(2*n+1,sqrt(2+x)) / sqrt(2+x), with Chebyshev polynomials R with coefficients given in A127672 (scaled T-polynomials).
R(n,x) is called C_n(x) in Abramowitz and Stegun's handbook, p. 778, 22.5.11.
P(n,x) = S(n,x)-S(n-1,x), n>=0, S(-1,x)=0, with the Chebyshev S-polynomials (see the coefficient triangle A049310).
O.g.f. for row polynomials: P(x,z):= sum(n>=0, P(n,x)*z^n ) = (1-z)/(1-x*z+z^2).
  (from the o.g.f. for R(2*n+1,x), n>=0, computed from the o.g.f. for the R-polynomials (2-x*z)/(1-x*z+z^2) (see A127672))
Proof of the Chebyshev connection from the o.g.f. for Riordan array property of this triangle (see the P. Barry comment above).
For the A- and Z-sequences of this Riordan array see a comment above. (End)
abs(T(n,k)) = A046854(n,k) = abs(A066170(n,k)) T(n,n-k) = A108299(n,k); abs(T(n,n-k)) = A065941(n,k). - Johannes W. Meijer, Aug 08 2011
From Wolfdieter Lang, Jul 31 2014: (Start)
Similar to the triangles A157751, A244419 and A180070 one can give for the row polynomials P(n,x) besides the usual three term recurrence another one needing only one recurrence step. This uses also a negative argument, namely P(n,x) = (-1)^(n-1)*(-1 + x/2)*P(n-1,-x) + (x/2)*P(n-1,x), n >= 1, P(0,x) = 1. Proof by computing the o.g.f. and comparing with the known one. This entails the alternative triangle recurrence T(n,k) = (-1)^(n-k)*T(n-1,k) + (1/2)*(1 + (-1)^(n-k))*T(n-1,k-1), n >= m >= 1, T(n,k) = 0 if n < k and T(n,0) = (-1)^floor((n+1)/2) = A057077(n+1). [P(n,x) recurrence corrected Aug 03 2014]
(End)

New name and Chebyshev comments by Wolfdieter Lang, Jun 11 2010

A244419 Coefficient triangle of polynomials related to the Dirichlet kernel. Rising powers. Riordan triangle ((1+z)/(1+z^2), 2*z/(1+z^2)).

Original entry on oeis.org

1, 1, 2, -1, 2, 4, -1, -4, 4, 8, 1, -4, -12, 8, 16, 1, 6, -12, -32, 16, 32, -1, 6, 24, -32, -80, 32, 64, -1, -8, 24, 80, -80, -192, 64, 128, 1, -8, -40, 80, 240, -192, -448, 128, 256, 1, 10, -40, -160, 240, 672, -448, -1024, 256, 512, -1, 10, 60, -160, -560, 672, 1792, -1024, -2304, 512, 1024

Offset: 0

Wolfdieter Lang, Jul 29 2014

This is the row reversed version of A180870. See also A157751 and A228565.
The Dirichlet kernel is D(n,x) = Sum_{k=-n..n} exp(i*k*x) = 1 + 2*Sum_{k=1..n} T(n,x) = S(n, 2*y) + S(n-1, 2*y) = S(2*n, sqrt(2*(1+y))) with y = cos(x), n >= 0, with the Chebyshev polynomials T (A053120) and S (A049310). This triangle T(n, k) gives in row n the coefficients of the polynomial Dir(n,y) = D(n,x=arccos(y)) = Sum_{m=0..n} T(n,m)*y^m. See A180870, especially the Peter Bala comments and formulas.
This is the Riordan triangle ((1+z)/(1+z^2), 2*z/(1+z^2)) due to the o.g.f. for Dir(n,y) given by (1+z)/(1 - 2*y*z + z^2) = G(z)/(1 - y*F(z)) with G(z) = (1+z)/(1+z^2) and F(z) = 2*z/(1+z^2) (see the Peter Bala formula under A180870). For Riordan triangles and references see the W. Lang link 'Sheffer a- and z- sequences' under A006232.
The A- and Z- sequences of this Riordan triangle are (see the mentioned W. Lang link in the preceding comment also for the references): The A-sequence has o.g.f. 1+sqrt(1-x^2) and is given by A(2*k+1) = 0 and A(2*k) [2, -1/2, -1/8, -1/16, -5/128, -7/256, -21/1024, -33/2048, -429/32768, -715/65536, ...], k >= 0. (See A098597 and A046161.)
  The Z-sequence has o.g.f. sqrt((1-x)/(1+x)) and is given by
  [1, -1, 1/2, -1/2, 3/8, -3/8, 5/16, -5/16, 35/128, -35/128, ...]. (See A001790 and A046161.)
The column sequences are A057077, 2*(A004526 with even numbers signed), 4*A008805 (signed), 8*A058187 (signed), 16*A189976 (signed), 32*A189980 (signed) for m = 0, 1, ..., 5.
The row sums give A005408 (from the o.g.f. due to the Riordan property), and the alternating row sums give A033999.
The row polynomials Dir(n, x), n >= 0, give solutions to the diophantine equation (a + 1)*X^2 - (a - 1)*Y^2 = 2 by virtue of the identity (a + 1)*Dir(n, -a)^2 - (a - 1)*Dir(n, a)^2 = 2, which is easily proved inductively using the recurrence Dir(n, a) = (1 + a)*(-1)^(n-1)*Dir(n-1, -a) + a*Dir(n-1, a) given below by Wolfdieter Lang. - Peter Bala, May 08 2025

			The triangle T(n,m) begins:
  n\m  0   1   2    3    4    5    6     7     8    9    10 ...
  0:   1
  1:   1   2
  2:  -1   2   4
  3:  -1  -4   4    8
  4:   1  -4 -12    8   16
  5:   1   6 -12  -32   16   32
  6:  -1   6  24  -32  -80   32   64
  7:  -1  -8  24   80  -80 -192   64   128
  8:   1  -8 -40   80  240 -192 -448   128   256
  9:   1  10 -40 -160  240  672 -448 -1024   256  512
  10: -1  10  60 -160 -560  672 1792 -1024 -2304  512  1024
  ...
Example for A-sequence recurrence: T(3,1) = Sum_{j=0..2} A(j)*T(2,j) = 2*(-1) + 0*2 + (-1/2)*4 = -4. Example for Z-sequence recurrence: T(4,0) = Sum_{j=0..3} Z(j)*T(3,j) = 1*(-1) + (-1)*(-4) + (1/2)*4 + (-1/2)*8 = +1. (For the A- and Z-sequences see a comment above.)
Example for the alternate recurrence: T(4,2) = 2*T(3,1) - T(3,2) = 2*(-4) - 4 = -12. T(4,3) = 0*T(3,2) + T(3,3) = T(3,3) = 8. - _Wolfdieter Lang_, Jul 30 2014

Wikipedia, Dirichlet kernel.

Dir(n, x) : A005408 (x = 1), A002878 (x = 3/2), A001834 (x = 2), A030221 (x = 5/2), A002315 (x = 3), A033890 (x = 7/2), A057080 (x = 4), A057081 (x = 9/2), A054320 (x = 5), A077416 (x = 6), A028230 (x = 7), A159678 (x = 8), A049629 (x = 9), A083043 (x = 10),
(-1)^n * Dir(n, x): A122367 (x = -3/2); A079935 (x = -2), A004253 (x = -5/2), A001653 (x = -3), A049685 (x = -7/2), A070997 (x = -4), A070998 (x = -9/2), A072256(n+1) (x = -5).
Cf. A180870 (row reversed), A157751, A228565, A049310, A053120, A057077, A004526, A008805, A058187, A189976, A189980, A005408, A033999.

T[n_, k_] := T[n, k] = Which[k == 0, (-1)^Quotient[n, 2], (0 <= n && n < k) || (n == -1 && k == 1), 0, True, 2 T[n-1, k-1] - T[n-2, k]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2019, from Sage *)

def T(n, k):
    if k == 0: return (-1)^(n//2)
    if (0 <= n and n < k) or (n == -1 and k == 1): return 0
    return 2*T(n-1, k-1) - T(n-2, k)
for n in range(11): [T(n,k) for k in (0..n)] # Peter Luschny, Jul 29 2014

T(n, m) = [y^m] Dir(n,y) for n >= m >= 0 and 0 otherwise, with the polynomials Dir(y) defined in a comment above.
T(n, m) = 2^m*(S(n,m) + S(n-1,m)) with the entries S(n,m) of A049310 given there explicitly.
O.g.f. for polynomials Dir(y) see a comment above (Riordan triangle ((1+z)/(1+z^2), 2*z/(1+z^2))).
O.g.f. for column m: ((1 + x)/(1 + x^2))*(2*x/(1 + x^2))^m, m >= 0, (Riordan property).
Recurrence for the polynomials: Dir(n, y) = 2*y*Dir(n-1, y) - Dir(n-2, y), n >= 1, with input D(-1, y) = -1 and D(0, y) = 1.
Triangle three-term recurrence: T(n,m) = 2*T(n-1,m-1) - T(n-2,m) for n >= m >= 1 with T(n,m) = 0 if 0 <= n < m, T(0,0) = 1, T(-1,1) = 0 and T(n,0) = A057077(n) = (-1)^(floor(n/2)).
From Wolfdieter Lang, Jul 30 2014: (Start)
In analogy to A157751 one can derive a recurrence for the row polynomials Dir(n, y) = Sum_{m=0..n} T(n,m)*y^m also using a negative argument but only one recursive step: Dir(n,y) = (1+y)*(-1)^(n-1)*Dir(n-1,-y) + y*Dir(n-1,y), n >= 1, Dir(0,y) = 1 (Dir(-1,y) = -1). See also A180870 from where this formula can be derived by row reversion.
This entails another triangle recurrence T(n,m) = (1 + (-1)^(n-m))*T(n-1,m-1) - (-1)^(n-m)*T(n-1,m), for n >= m >= 1 with T(n,m) = 0 if n < m and T(n,0) = (-1)^floor(n/2). (End)
From Peter Bala, Aug 14 2022: (Start)
The row polynomials Dir(n,x), n >= 0, are related to the Chebyshev polynomials of the first kind T(n,x) by the binomial transform as follows:
(2^n)*(x - 1)^(n+1)*Dir(n,x) = (-1) * Sum_{k = 0..2*n+1} binomial(2*n+1,k)*T(k,-x).
Note that Sum_{k = 0..2*n} binomial(2*n,k)*T(k,x) = (2^n)*(1 + x)^n*T(n,x). (End)
From Peter Bala, May 04 2025: (Start)
For n >= 1, the n-th row polynomial Dir(n, x) = (-1)^n * (U(n, -x) - U(n-1, -x)) = U(2*n, sqrt((1+x)/2)), where U(n, x) denotes the n-th Chebyshev polynomial of the second kind.
For n >= 1 and x < 1, Dir(n, x) = (-1)^n * sqrt(2/(1 - x )) * T(2*n+1, sqrt((1 - x)/2)), where T(n, x) denotes the n-th Chebyshev polynomial of the first kind.
Dir(n, x)^2 - 2*x*Dir(n, x)*Dir(n+1, x) + Dir(n+1, x)^2 = 2*(1 + x).
Dir(n, x) = (-1)^n * R(n, -2*(x+1)), where R(n, x) is the n-th row polynomial of the triangle A085478.
Dir(n, x) = Sum_{k = 0..n} (-1)^(n+k) * binomial(n+k, 2*k) * (2*x + 2)^k. (End)

A180870 D(n, x) is the Dirichlet kernel sin((n+1/2)x)/sin(x/2). The triangle gives in row n the coefficients of descending powers of x of the polynomial D(n, arccos(x)).

Original entry on oeis.org

1, 2, 1, 4, 2, -1, 8, 4, -4, -1, 16, 8, -12, -4, 1, 32, 16, -32, -12, 6, 1, 64, 32, -80, -32, 24, 6, -1, 128, 64, -192, -80, 80, 24, -8, -1, 256, 128, -448, -192, 240, 80, -40, -8, 1, 512, 256, -1024, -448, 672, 240, -160, -40, 10, 1

Offset: 0

Jonny Griffiths, Sep 21 2010

D(n, arccos(x)) = U(n, x) + U(n-1, x) where U(n, x) are the Chebyshev polynomials of the second kind. These polynomials arise naturally in the investigation of the integer triples (p, q, (p*q + 1)/(p + q)).

Chebyshev polynomials of the fourth kind, usually denoted by W(n, x) (see, for example, Mason and Handscomb, Chapter 1, Definition 1.3). See A228565 for Chebyshev polynomials of the third kind. Cf. A157751. - Peter Bala, Jan 17 2014

			The triangle T(n,m) begins:
n\m    0   1     2     3    4   5    6    7   8  9  10 ...
0:     1
1:     2   1
2:     4   2    -1
3:     8   4    -4    -1
4:    16   8   -12    -4    1
5:    32  16   -32   -12    6   1
6:    64  32   -80   -32   24   6   -1
7:   128  64  -192   -80   80  24   -8   -1
8:   256 128  -448  -192  240  80  -40   -8   1
9:   512 256 -1024  -448  672 240 -160  -40  10  1
10: 1024 512 -2304 -1024 1792 672 -560 -160  60 10  -1
... reformatted - _Wolfdieter Lang_, Jul 26 2014
Recurrence: T(4,2) = (1 + 1)*T(3,2) - T(3,1) = 2*(-4) - 4 = -12. T(4,3) = 0*T(3,3) - (-1)*T(3,2) = T(3,2) = -4. - _Wolfdieter Lang_, Jul 30 2014

J. C. Mason and D. C. Handscomb, Chebyshev polynomials, Chapman and Hall/CRC 2002.

Paul Barry, On the Group of Almost-Riordan Arrays, arXiv preprint arXiv:1606.05077 [math.CO], 2016.
Wikipedia, Dirichlet kernel.

Cf. A008312, A028297, A157751, A228565, A049310, A244419 (row reversed triangle).

ogf := (1 + t)/(1 - 2*x*t + t^2):
ser := simplify(series(ogf, t, 12)): tc := n -> coeff(ser, t, n):
Trow := n -> local k; seq(coeff(tc(n), x, n-k), k = 0..n):
seq(print(Trow(n)), n = 0..9);  # Peter Luschny, Oct 07 2024

row(n) = {if (n==0, return([1])); f = 2*x+1; for (k = 2, n, for (i = 1, (k-1)\2 + 1, f += (-1)^(i+1)*(binomial(k-i, i-1)*(2*x)^(k-2*i+2) - 2*binomial(k-1-i, i-1)*(2*x)^(k-2*i)););); Vec(f);} \\ Michel Marcus, Jul 18 2014

From Peter Bala, Jan 17 2014: (Start)
O.g.f. (1 + t)/(1 - 2*x*t + t^2) = 1 + (2*x + 1)*t + (4*x^2 + 2*x - 1)*t^2 + ....
Recurrence equation: W(0,x) = 1, W(1,x) = 2*x + 1 and W(n,x) = 2*x*W(n-1,x) - W(n-2,x) for n >= 2.
In terms of U(n,x), the Chebyshev polynomials of the second kind, we have W(n,x) = U(2*n,u) with u = sqrt((1 + x)/2). Also binomial(2*n,n)*W(n,x) = 2^(2*n)*Jacobi_P(n,1/2,-1/2,x). (End)
Row sums: 2*n+1. - Michel Marcus, Jul 16 2014
T(n,m) = [x^(n-m)](U(n, x) + U(n-1, x)) = [x^(n-m)] S(2*n, sqrt(2*(1+x))), n >= m >= 0, with U(n, x) = S(n, 2*x). The coefficient triangle of the Chebyshev S-polynomials is given in A049310. See the Peter Bala comments above. - Wolfdieter Lang, Jul 26 2014
From Wolfdieter Lang, Jul 30 2014: (Start)
O.g.f. for the row polynomials R(n,x) = Sum_{m=0..n} T(n,m)*x^m, obtained from the one given by Peter Bala above by row reversion: (1 + x*t)/(1 - 2*t + (x*t)^2).
In analogy to A157751 one can derive a recurrence for the row polynomials R(n, x) = x^n*Dir(n,1/x) with Dir(n,x) = U(n,x) + U(n-1,x) using also negative arguments but only one recursive step: R(n,x) = (1+x)*R(n-1,-x) + R(n-1,x), n >= 1, R(0,x) = 1 (R(-1,x) = -1/x). Proof: derive the o.g.f. and compare it with the known one.
This entails the triangle recurrence T(n,m) = (1 + (-1)^m)* T(n-1,m) - (-1)^m*T(n-1,m-1), for n >= m >= 1 with T(n,m) = 0 if n < m and T(n,0) = 2^n. (End)

Missing term in sequence corrected by Paul Curtz, Dec 31 2011

Edited (name reformulated, Wikipedia link added) by Wolfdieter Lang, Jul 26 2014

A140070 Triangle read by rows, iterates of matrix X * [1,0,0,0,...], where X = an infinite lower bidiagonal matrix with [1,3,1,3,1,3,...] in the main diagonal and [1,1,1,...] in the subdiagonal.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 13, 5, 1, 1, 40, 18, 8, 1, 1, 121, 58, 42, 9, 1, 1, 364, 179, 184, 51, 12, 1, 1, 1093, 543, 731, 235, 87, 13, 1, 1, 3280, 1636, 2736, 966, 496, 100, 16, 1, 1, 9841, 4916, 9844, 3702, 2454, 596, 148, 17, 1, 1, 29524, 14757, 34448, 13546, 11064, 3050, 1040, 165, 20, 1

Offset: 1

Gary W. Adamson, May 04 2008

Row sums = A006012: (1, 2, 6, 20, 68, 232, 792, 2704,...).

Companion to triangle A140071.

			First few rows of the triangle are:
  1;
  1,    1;
  1,    4,    1;
  1,   13,    5,    1;
  1,   40,   18,    8,   1;
  1,  121,   58,   42,   9,   1;
  1,  364,  179,  184,  51,  12,   1;
  1, 1093,  543,  731, 235,  87,  13,  1;
  1, 3280, 1636, 2736, 966, 496, 100, 16, 1;
  ...

Cf. A006012, A140071, A157751.

T:= proc(n, k) option remember;
      `if`(k<0 or k>n, 0, `if`(k=0 or k=n, 1,
      4*T(n-1, k) - 3*T(n-2, k) + T(n-2, k-2)))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Feb 18 2020

With[{m = 10}, CoefficientList[CoefficientList[Series[(1 + (y - 3)*x)/(1 - 4*x - (y^2 - 3)*x^2), {x, 0, m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 18 2020 *)

Triangle read by rows, iterates of matrix X * [1,0,0,0,...], where X = an infinite lower bidiagonal matrix with [1,3,1,3,1,3,...] in the main diagonal and [1,1,1,...] in the subdiagonal; with the rest zeros.
From Peter Bala, Jan 17 2014: (Start)
O.g.f.: (1 + (x - 3)*z)/(1 - 4*z - (x^2 - 3)*z^2) = 1 + (x + 1)*z + (x^2 + 4*x + 1)*z^2 + ....
Recurrence equation: T(n,k) = 4*T(n-1,k) - 3*T(n-2,k) + T(n-2,k-2).
Recurrence equation for row polynomials: R(n,x) = 4*R(n-1,x) + (x^2 - 3)*R(n-2,x) with R(0,x) = 1 and R(1,x) = 1 + x.
Another recurrence equation: R(n,x) = (x + 2)*R(n-1,x) - R(n-1,-x) with R(0,x) = 1. Cf. A157751. (End)

A140071 Triangle read by rows: iterates of X * [1,0,0,0,...]; where X = an infinite lower bidiagonal matrix with [3,1,3,1,3,1...] in the main diagonal and [1,1,1,...] in the subdiagonal.

Original entry on oeis.org

1, 3, 1, 9, 4, 1, 27, 13, 7, 1, 81, 40, 34, 8, 1, 243, 121, 142, 42, 11, 1, 729, 364, 547, 184, 75, 12, 1, 2187, 1093, 2005, 731, 409, 87, 15, 1, 6561, 3280, 7108, 2736, 1958, 496, 132, 16, 1, 19683, 9841, 24604, 9844, 8610, 2454, 892, 148, 19, 1

Offset: 1

Gary W. Adamson and Roger L. Bagula, May 04 2008

Companion triangle A140070 uses an analogous operation with the main diagonal [1,3,1,3,1,3,...].

			First few rows of the triangle are:
1;
3, 1;
9, 4, 1;
27, 13, 7, 1;
81, 40, 34, 8, 1;
243, 121, 142, 42, 11, 1;
729, 364, 547, 184, 75, 12, 1;
2187, 1093, 2005, 731, 409, 87, 15, 1;
6561, 3280, 7108, 2736, 1958, 496, 132, 16, 1;
...

Cf. A140070, A007070 (row sums), A157751.

From Peter Bala, Jan 17 2014: (Start)
O.g.f. (1 + (x - 1)*z)/(1 - 4*z - (x^2 - 3)*z^2) = 1 + (x + 3)*z + (x^2 + 4*x + 9)*z^2 + ....
Recurrence equation: T(n,k) = 4*T(n-1,k) - 3*T(n-2,k) + T(n-2,k-2).
Recurrence equation for row polynomials: R(n,x) = 4*R(n-1,x) + (x^2 - 3)*R(n-2,x) with R(0,x) = 1 and R(1,x) = 3 + x.
Another recurrence equation: R(n,x) = (x + 2)*R(n-1,x) + R(n-1,-x) with R(0,x) = 1. Cf. A157751. (End)

cp's OEIS Frontend

A192313 Constant term of the reduction of n-th polynomial at A157751 by x^2->x+1.

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A192338 Constant term of the reduction of n-th polynomial at A157751 by x^2->x+2.

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A192337 Coefficient of x in the reduction of n-th polynomial at A157751 by x^2->x+1.

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A192339 Coefficient of x in the reduction of n-th polynomial at A157751 by x^2->x+2.

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A130777 Coefficients of first difference of Chebyshev S polynomials.

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A244419 Coefficient triangle of polynomials related to the Dirichlet kernel. Rising powers. Riordan triangle ((1+z)/(1+z^2), 2*z/(1+z^2)).

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A180870 D(n, x) is the Dirichlet kernel sin((n+1/2)x)/sin(x/2). The triangle gives in row n the coefficients of descending powers of x of the polynomial D(n, arccos(x)).

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