A028297 -id:A028297 - cp's OEIS frontend

A288188 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=8 data values.

1, 8, -7, 64, -84, 21, 512, -896, 224, 196, -35, 4096, -8960, 2240, 3920, -350, -980, 35, 32768, -86016, 21504, 56448, -3360, -18816, 336, -5488, 1470, 1176, -21, 262144, -802816, 200704, 702464, -31360, -263424, 3136, -153664, 27440, 21952, -196, 38416, -1372, -3430, 7

Offset: 1

Gregory Gerard Wojnar, Jun 16 2017

Let SM_k = Sum( d_(t_1, t_2, t_3, ..., t_8)* eM_1^t_1 * eM_2^t_2 * ...*eM_8^t_8) summed over all length 8 integer partitions of k, i.e., 1*t_1+2*t_2+3*t_3+...+8*t_8=k, where SM_k are the averaged k-th power sum symmetric polynomials in 8 data (i.e., SM_k = S_k/8 where S_k are the k-th power sum symmetric polynomials, and where eM_k are the averaged k-th elementary symmetric polynomials, eM_k = e_k/binomial(8,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2, t_3, ..., t_8) form a triangle, with one row for each k value starting with k=1; the number of terms in successive rows is nondecreasing.

Row sums of positive entries give: 1,8,85,932,10291,114878,... Row sums of negative entries are always 1 less than corresponding row sums of positive entries.

			Triangle begins
     1;
     8,    -7;
    64,   -84,   21;
   512,  -896,  224,  196,  -35;
  4096, -8960, 2240, 3920, -350, -980, 35;
  ...

Gregory Gerard Wojnar, Java program
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, pp. 22-24, arXiv:1706.08381 [math.GM], 2017.

Cf. A028297 (m=2), A287768 (m=3), A288199 (m=4), A288207 (m=5), A288211 (m=6), A288245 (m=7). See Girard-Waring A210258. T(n,1)=8^(n-1)=A001018(n).

Java
```
// See Wojnar link.
```

A288199 Irregular triangle read by rows: mean version of Girard-Waring formula (cf. A210258), for m = 4 data values.

Original entry on oeis.org

1, 4, -3, 16, -18, 3, 64, -96, 16, 18, -1, 256, -480, 80, 180, -30, -5, 1024, -2304, 384, 1296, -288, -108, -24, 9, 12, 4096, -10752, 1792, 8064, -2016, -1512, 112, 252, -112, 84, -7

Offset: 1

Gregory Gerard Wojnar, May 31 2017

Let SM_k = Sum( d_(t_1, t_2, t_3, t_4)* eM_1^t_1 * eM_2^t_2 * eM_3^t_3*eM_4^t_4) summed over all length 4 integer partitions of k, i.e., 1*t_1 + 2*t_2 + 3*t_3 + 4*t_4 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 4 data (i.e., SM_k = S_k/4 where S_k are the k-th power sum symmetric polynomials, and where eM_k are the averaged k-th elementary symmetric polynomials, eM_k = e_k/binomial(4,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2, t_3, t_4) form an irregular triangle, with one row for each k value starting with k=1; "irregular" means that the number of terms in successive rows is nondecreasing.
Row sums of positive entries give 1, 4, 19, 98, 516, 2725, 14400.
Row sums of negative entries are always 1 less than corresponding row sums of positive entries.

			Triangle begins:
    1;
    4,   -3;
   16,  -18,   3;
   64,  -96,  16,  18, -1;
  256, -480,  80, 180, -5, -30;
  ...
The first few rows describe:
Row 1: SM_1 = 1 eM_1;
Row 2: SM_2 = 4*(eM_1)^2 - 3*eM_2;
Row 3: SM_3 = 16*(eM_1)^3 - 18*eM_1*eM_2 + 3*eM_3;
Row 4: SM_4 = 64*(eM_1)^4 - 96*(eM_1)^2*eM_2 + 16*eM_1*eM_3 + 18*(eM_2)^2 - 1*eM_4;
Row 5: SM_5 = 256*(eM_1)^5 - 480*(eM_1)^3*eM_2 + 80*(eM_1)^2*eM_2 + 180*eM_1*(eM_2)^2 - 30*eM_2*eM_3 - 5*eM_1*eM_4.

Gregory Gerard Wojnar, Java program
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, Table GW.n=4, p.23, arXiv:1706.08381 [math.GM], 2017.

Cf. A210258, A028297 (m=2), A287768 (m=3), A288207 (m=5), A288211 (m=6), A288245 (m=7), A288188 (m=8).

A288211 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=6 data values.

Original entry on oeis.org

1, 6, -5, 36, -45, 10, 216, -360, 80, 75, -10, 1296, -2700, 600, 1125, -250, -75, 5, 7776, -19440, 4320, 12150, -3600, -1125, -540, 225, 200, 36, -1, 46656, -136080, 30240, 113400, -37800, -23625, 2800, 5250, -3780, 3150, -350, 252, -105, -7

Offset: 1

Gregory Gerard Wojnar, Jun 06 2017

Let SM_k = Sum( d_(t_1, t_2, ..., t_6)* eM_1^t_1 * eM_2^t_2 * ... * eM_6^t_6) summed over all length 6 integer partitions of k, i.e., 1*t_1 + 2*t_2 + 3*t_3 + ... + 6*t_6 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 6 data (i.e., SM_k = S_k/6 where S_k are the k-th power sum symmetric polynomials, and where eM_k are the averaged k-th elementary symmetric polynomials, eM_k = e_k/binomial(6,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2, t_3, ..., t_6) form a triangle, with one row for each k value starting with k=1; the number of terms in successive rows is nondecreasing.

Row sums of positive entries give 1,6,46,371,3026,24707,201748. Row sums of negative entries are always 1 less than corresponding row sums of positive entries.

			Triangle begins:
 1;
 6,-5;
 36,-45,10;
 216,-360,80,75,-10;
 1296,-2700,600,1125,-250,-75,5;
 7776,-19440,4320,12150,-3600,-1125,-540,225,200,36,-1;
 ...
Above represents:
 SM_1 = eM_1;
 SM_2 = 6*(eM_1)^2 - 5*eM_2;
 SM_3 = 36*(eM_1)^3 - 45*eM_1*eM_2 + 10*eM_3;
 SM_4 = 216*(eM_1)^4 - 360*(eM_1)^2*eM_2 + 80*eM_1*eM_3 + 75*(eM_2)^2 - 10*eM_4;
 SM_5 = 1296*(eM_1)^5 - 2700*(eM_1)^3*eM_2 + 600*(eM_1)^2*eM_3 + 1125*eM_1*(eM_2)^2 - 250*eM_2*eM_3 - 75*eM_1*eM_4 + 5*eM_5;
 ...

Gregory Gerard Wojnar, Java program
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, Table GW.n=6, p.24, arXiv:1706.08381 [math.GM], 2017.

Cf. A028297 (m=2), A287768 (m=3), A288199 (m=4), A288207 (m=5), A288245 (m=7), A288188 (m=8). Also see A210258 Girard-Waring.

First column of triangle is powers of m=6, A000400.

Java
```
// See link.
```

A288245 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=7 data values.

Original entry on oeis.org

1, 7, -6, 49, -63, 15, 343, -588, 140, 126, -20, 2401, -5145, 1225, 2205, -175, -525, 15, 16807, -43218, 10290, 27783, -1470, -8820, 126, -2646, 630, 525, -6, 117649, -352947, 84035, 302526, -12005, -108045, 1029, -64827, 10290, 8575, -49, 15435, -441, -1225, 1

Offset: 1

Gregory Gerard Wojnar, Jun 06 2017

Let SM_k = Sum( d_(t_1, t_2, t_3, ..., t_7)* eM_1^t_1 * eM_2^t_2 * ... * eM_7^t_7) summed over all length 7 integer partitions of k, i.e., 1*t_1 + 2*t_2 + 3*t_3 + ... + 7*t_7 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 7 data (i.e., SM_k = S_k/7 where S_k are the k-th power sum symmetric polynomials, and where eM_k are the averaged k-th elementary symmetric polynomials, eM_k = e_k/binomial(7,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2, t_3, ..., t_7) form a triangle, with one row for each k value starting with k=1; the number of terms in successive rows is nondecreasing.

Row sums of positive entries give 1,7,64,609,5846,56161,... Row sums of negative entries are always 1 less than corresponding row sums of positive entries.

			Triangular array begins...
1;
7,-6;
49,-63,15;
343,-588,140,126,-20;
2401,-5145,1225,2205,-175,-525,15;
16807,-43218,10290,27783,-1470,-8820,126,-2646,630,525,-6;
117649,-352947,84035,302526,-12005,-108045,1029,64827,10290,8575,-49,15435,-441,-1225,1;

Gregory Gerard Wojnar, Java program
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, pp. 22-24, arXiv:1706.08381 [math.GM], 2107.

Cf. A028297 (m=2), A287768 (m=3), A288199 (m=4), A288211 (m=5), A288211 (m=6), A288188 (m=8). Also see Girard-Waring A210258.

First entries of each row of triangle are powers of m=7, A000420.

Java
```
// See Wojnar link.
```

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

A228565 Triangle read by rows: coefficients of descending powers of the polynomial V(n,x) = cos((2n+1)(arccos(x)/2))/cos(arccos(x)/2), n >= 0.

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, 1024, -512, -2304, 1024, 1792, -672, -560, 160, 60, -10, -1, 2048, -1024, -5120, 2304, 4608, -1792, -1792, 560, 280, -60, -12, 1, 4096, -2048, -11264, 5120, 11520, -4608, -5376, 1792, 1120, -280, -84, 12, 1

Offset: 0

Jonny Griffiths, Aug 25 2013

V(n,x) is related to the Dirichlet kernel and its associated polynomials. V(n,x) arises in studying recurrences connecting the Chebyshev polynomials of the first and second kinds. It differs from A180870 above only in the signs of terms.

Chebyshev polynomials V(n,x) of the third kind (see, for example, Mason and Handscomb, Chapter 1, Definition 1.3). See A180870 for Chebyshev polynomials of the fourth kind. Cf. A155751. - Peter Bala, Jan 17 2014

			V(0,x) = 1, V(1,x) = 2x-1, V(2,x) = 4x^2-2x-1, V(3,x) = 8x^3 -4x^2 - 4x + 1, V(4,x) = 16x^4 - 8x^3 - 12x^2 + 4x + 1, V(5,x) = 32x^5 - 16x^4 - 32x^3 + 12x^2 + 6x - 1, V(6,x) =64x^6 - 32x^5 - 80x^4 + 32x^3 + 24x^2 - 6x - 1, ...
Triangle begins:
     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;
  1024, -512, -2304, 1024, 1792, -672, -560, 160,  60, -10,  -1;
  ...

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.

Cf. A180870, A028297. A008312, A155751.

A228565 := proc(n,k)
    local t,Vn,x ;
    t := arccos(x) ;
    Vn := cos((n+1/2)*t)/cos(t/2) ;
    coeftayl(%,x=0,n-k) ;
end proc:
for n from 0 to 10 do
    for k from 0 to n do
        printf("%d,",A228565(n,k)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Mar 12 2014

V[n_] := Cos[(2*n + 1)*(ArcCos[x]/2)]/Cos[ArcCos[x]/2];
row[n_] := CoefficientList[V[n] + O[x]^(n + 1), x] // Reverse;
Table[row[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Nov 20 2017 *)

V(n+1,x) = 2xV(n,x) - V(n-1,x) with V(0,x) = 1, V(1,x) = 2x-1.
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 + ....
In terms of the Chebyshev polynomials T(n,x) of the first kind and Chebyshev polynomials U(n,x) of the second kind we have
V(n,x) = U(n,x) - U(n-1,x);
V(n,x) + V(n-1,x) = 2*T(n,x);
V(n,x) = 1/u*T(2*n+1,u) with u = sqrt((1 + x)/2).
Also binomial(2*n,n)*V(n,x) = 2^(2*n)*Jacobi_P(n,-1/2,1/2,x). (End)

A288207 Irregular triangle read by rows: mean version of Girard-Waring formula A210258, for m = 5 data values.

Original entry on oeis.org

1, 5, -4, 25, -30, 6, 125, -200, 40, 40, -4, 625, -1250, 250, 500, -25, -100, 1, 3125, -7500, 1500, 4500, -150, -1200, 6, -400, 60, 60, 15625, -43750, 8750, 35000, -875, -10500, 35, -7000, 700, 700, 1400, -14, -70

Offset: 1

Gregory Gerard Wojnar, Jun 06 2017

Let SM_k = Sum( d_(t_1, t_2, ... , t_5)* eM_1^t_1 * eM_2^t_2 *...* eM_5^t_5) summed over all length 5 integer partitions of k, i.e., 1*t_1+2*t_2+...+5*t_5=k, where SM_k are the averaged k-th power sum symmetric polynomials in 5 data (i.e., SM_k = S_k/5 where S_k are the k-th power sum symmetric polynomials, and where eM_k are the averaged k-th elementary symmetric polynomials, eM_k = e_k/binomial(5,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2,... , t_5) form a triangle, with one row for each k value starting with k=1; the number of terms in successive rows is nondecreasing.

			Triangle begins:
    1;
    5,    -4;
   25,   -30,   6;
  125,  -200,  40,  40,   -4;
  625, -1250, 250, 500, -100, -25, 1;
  ...
Above represents:
SM_1 = 1*eM_1;
SM_2 = 5*(eM_1)^2 -4*eM_2;
SM_3 = 25*(eM_1)^3 - 30*eM_1*eM_2 + 6*eM_3;
SM_4 = 125*(eM_1)^4 - 200*(eM_1)^2*eM_2 + 40*eM_1*eM_3 + 40*(eM_2)^2 - 4*eM_4;
SM_5 = 625*(eM_1)^5 - 1250*(eM_1)^3*eM_2 + 250*(eM_1)^2*eM_3 + 500*eM_1*(eM_2)^2 - 100*eM_2*eM_3 - 25*eM_1*eM_4 + 1*eM_5;
...

Gregory Gerard Wojnar, Java program

Cf. A028297 (m=2), A287768 (m=3), A288199 (m=4), A288211 (m=6), A288245 (m=7), A288188 (m=8); A210258 Girard-Waring.

First column of triangle are powers of m=5, A000351.

Java
```
// See Java program link.
```

A317405 a(n) = n * A001353(n).

Original entry on oeis.org

1, 8, 45, 224, 1045, 4680, 20377, 86912, 364905, 1513160, 6211909, 25290720, 102251773, 410963336, 1643288625, 6541692416, 25939798993, 102503274120, 403800061789, 1586318259680, 6216231359205, 24304019419592, 94826736906697, 369285078314880, 1435615286196025

Offset: 1

Rigoberto Florez, Jul 27 2018

Derivative of Chebyshev polynomials of the first kind evaluated at x=2.

Colin Barker, Table of n, a(n) for n = 1..1000
Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6.
R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the First Kind
Index entries for linear recurrences with constant coefficients, signature (8,-18,8,-1).

Cf. A001353, A028297 (Chebyshev polynomials of the first kind).

Table[ D[ ChebyshevT[n, x], x] /. x -> 2, {n, 25}]
CoefficientList[Series[-x(x^2 - 1)/(x^2 - 4x + 1)^2, {x, 0, 24}], x] (* Robert G. Wilson v, Aug 07 2018 *)

Vec(x*(1 - x)*(1 + x) / (1 - 4*x + x^2)^2 + O(x^40)) \\ Colin Barker, Jul 28 2018

a(n) = subst(deriv(polchebyshev(n)), x, 2); \\ Michel Marcus, Jul 29 2018

From Colin Barker, Jul 28 2018: (Start)
G.f.: x*(1 - x)*(1 + x) / (1 - 4*x + x^2)^2.
a(n) = (((-(2-sqrt(3))^n + (2+sqrt(3))^n)*n)) / (2*sqrt(3)).
a(n) = 8*a(n-1) - 18*a(n-2) + 8*a(n-3) - a(n-4) for n>4.
(End)

A207537 Triangle of coefficients of polynomials u(n,x) jointly generated with A207538; see Formula section.

Original entry on oeis.org

1, 2, 1, 4, 3, 8, 8, 1, 16, 20, 5, 32, 48, 18, 1, 64, 112, 56, 7, 128, 256, 160, 32, 1, 256, 576, 432, 120, 9, 512, 1280, 1120, 400, 50, 1, 1024, 2816, 2816, 1232, 220, 11, 2048, 6144, 6912, 3584, 840, 72, 1, 4096, 13312, 16640, 9984, 2912, 364, 13

Offset: 1

Clark Kimberling, Feb 18 2012

Another version in A201701. - Philippe Deléham, Mar 03 2012
Subtriangle of the triangle given by (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 03 2012
Columns: A011782, A001792, A001793, A001794, A006974, A006975, A006976. - Philippe Deléham, Mar 03 2012
Diagonal sums: A052980. - Philippe Deléham, Mar 03 2012

			First seven rows:
   1;
   2,   1;
   4,   3;
   8,   8,  1;
  16,  20,  5,
  32,  48, 18, 1;
  64, 112, 56, 7;
From _Philippe Deléham_, Mar 03 2012: (Start)
Triangle A201701 begins:
   1;
   1,   0;
   2,   1,  0;
   4,   3,  0, 0;
   8,   8,  1, 0, 0;
  16,  20,  5, 0, 0, 0;
  32,  48, 18, 1, 0, 0, 0;
  64, 112, 56, 7, 0, 0, 0, 0;
  ... (End)

Cf. A028297, A207538, A133156.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x]
v[n_, x_] := u[n - 1, x] + v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207537, |A028297| *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A207538, |A133156| *)
(* Prepending 1 and with offset 0: *)
Tpoly[n_] := HypergeometricPFQ[{-n/2, -n/2 + 1/2}, {1/2}, x + 1];
Table[CoefficientList[Tpoly[n], x], {n, 0, 12}] // Flatten (* Peter Luschny, Feb 03 2021 *)

u(n,x) = u(n-1,x) + (x+1)*v(n-1,x), v(n,x) = u(n-1,x) + v(n-1,x), where u(1,x)=1, v(1,x)=1.  Also, A207537 = |A028297|.
T(n,k) = 2*T(n-1,k) + T(n-2,k-1). - Philippe Deléham, Mar 03 2012
G.f.: -(1+x*y)*x*y/(-1+2*x+x^2*y). - R. J. Mathar, Aug 11 2015
T(n, k) = [x^k] hypergeom([-n/2, -n/2 + 1/2], [1/2], x + 1) provided offset is set to 0 and 1 prepended. - Peter Luschny, Feb 03 2021

A123956 Triangle of coefficients of polynomials P(k) = 2XP(k - 1) - P(k - 2), P(0) = -1, P(1) = 1 + X, with twisted signs.

Original entry on oeis.org

-1, 1, 1, -1, -2, -2, 1, -3, 4, 4, -1, 4, 8, -8, -8, 1, 5, -12, -20, 16, 16, -1, -6, -18, 32, 48, -32, -32, 1, -7, 24, 56, -80, -112, 64, 64, -1, 8, 32, -80, -160, 192, 256, -128, -128, 1, 9, -40, -120, 240, 432, -448, -576, 256, 256, -1, -10, -50, 160, 400, -672, -1120, 1024, 1280, -512, -512

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 27 2006

Up to signs also the coefficients of polynomials y(n+1) = y(n-1) - 2*h*y(n), arising when the ODE y' = -y is numerically solved with the leapfrog (a.k.a. two-step Nyström) method, with y(0) = 1, y(1) = 1 - h. In this case, the coefficients are negative exactly for the odd powers of h. - M. F. Hasler, Nov 30 2022

			Triangle begins:
  {-1},
  { 1,   1},
  {-1,  -2,  -2},
  { 1,  -3,   4,    4},
  {-1,   4,   8,   -8,   -8},
  { 1,   5, -12,  -20,   16,   16},
  {-1,  -6, -18,   32,   48,  -32,   -32},
  { 1,  -7,  24,   56,  -80, -112,    64,   64},
  {-1,   8,  32,  -80, -160,  192,   256, -128, -128},
  { 1,   9, -40, -120,  240,  432,  -448, -576,  256,  256},
  {-1, -10, -50,  160,  400, -672, -1120, 1024, 1280, -512, -512},
  ...

CRC Standard Mathematical Tables and Formulae, 16th ed. 1996, p. 484.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Lutterbach, Approximating ODE y' = f(t,y) by using leapfrog method, Mathematics Stack Exchange, Nov 21 2019
Alastair MacDougall, 83.31 A Pascal-like triangle for coefficients of Chebyshev polynomials">, The Mathematical Gazette, Vol. 83, Issue 497 (Jul 1999), pp. 276-280.

Cf. A008312, A028297, A053117, A123235.

p[ -1, x] = 0; p[0, x] = 1; p[1, x] = x + 1;
p[k_, x_] := p[k, x] = 2*x*p[k - 1, x] - p[k - 2, x];
w = Table[CoefficientList[p[n, x], x], {n, 0, 10}];
An[d_] := Table[If[n == d && m  1/y)] /. 1/y -> 1, y], {d, 1, 11}] // Flatten

P=List([-1,1-'x]); {A123956(n,k)=for(i=#P, n+1, listput(P, P[i-1]-2*'x*P[i])); polcoef(P[n+1],k)*(-1)^((n-k-1)\2+!k*n\2)} \\ M. F. Hasler, Nov 30 2022

From M. F. Hasler, Nov 30 2022: (Start)
a(n,0) = (-1)^(n+1), a(n,1) = (-1)^floor(n/2)*n,
a(n,2) = (-1)^floor((n+1)/2)*A007590(n) = (-1)^floor((n+1)/2)*floor(n^2 / 2),
a(n,n) = a(n,n-1) = (-2)^(n-1) (n > 0),
a(n,3) / a(n,2) = { n/3 if n odd, -4*(n+2)/n if n even },
a(n,4) / a(n,3) = n/4 if n is even. (End)
From Peter Bala, Feb 06 2025: (Start)
Let T(n, x) and U(n, x) denote the n-th Chebyshev polynomial of the first and second kind. It appears that the row g.f.'s are as follows: for n >= 0,
row 4*n+1: T(4*n+1, x) + U(4*n, x); row 4*n+2: - 2 - T(4*n+2, x) - U(4*n+1, x);
row 4*n+3: 2 + T(4*n+3, x) + U(4*n+2, x); row 4*n+4: - T(4*n+4, x) - U(4*n+3, x). (End)

Offset changed to 0 by M. F. Hasler, Nov 30 2022

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