A267482 -id:A267482 - cp's OEIS frontend

A187660 Triangle read by rows: T(n,k) = (-1)^(floor(3k/2))binomial(floor((n+k)/2),k), 0 <= k <= n.

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

L. Edson Jeffery, Mar 12 2011

Conjecture: (i) Let n > 1 and N=2*n+1. Row n of T gives the coefficients of the characteristic polynomial p_N(x)=Sum_{k=0..n} T(n,k)*x^(n-k) of the n X n Danzer matrix D_{N,n-1} = {{0,...,0,1}, {0,...,0,1,1}, ..., {0,1,...,1}, {1,...,1}}. (ii) Let S_0(t)=1, S_1(t)=t and S_r(t)=t*S_(r-1)(t)-S_(r-2)(t), r > 1 (cf. A049310). Then p_N(x)=0 has solutions w_{N,j}=S_(n-1)(phi_{N,j}), where phi_{N,j}=2*(-1)^(j+1)*cos(j*Pi/N), j = 1..n. - L. Edson Jeffery, Dec 18 2011

			Triangle begins:
  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;

L. E. Jeffery, Danzer matrices
Guoce Xin and Yueming Zhong, Proving some conjectures on Kekulé numbers for certain benzenoids by using Chebyshev polynomials, arXiv:2201.02376 [math.CO], 2022.

Signed version of A046854.
Absolute values of a(n) form a reflected version of A065941, which is considered the main entry.
Cf. A046854, A066170, A130777, A267482.

A187660 := proc(n,k): (-1)^(floor(3*k/2))*binomial(floor((n+k)/2),k) end: seq(seq(A187660(n,k), k=0..n), n=0..11); # Johannes W. Meijer, Aug 08 2011

t[n_, k_] := (-1)^Floor[3 k/2] Binomial[Floor[(n + k)/2], k]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] (* L. Edson Jeffery, Oct 20 2017 *)

T(n,k) = (-1)^n*A066170(n,k).
abs(T(n,k)) = A046854(n,k) = abs(A066170(n,k)) = abs(A130777(n,k)).
abs(T(n,k)) = A065941(n,n-k) = abs(A108299(n,n-k)).

Edited and corrected by L. Edson Jeffery, Oct 20 2017

A267120 Triangle of coefficients of Gaussian polynomials [2n+3,3]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=3n.

Original entry on oeis.org

1, 0, -1, 1, 1, -1, 0, 5, -2, -4, 1, 1, 0, 2, -2, -15, 7, 17, -5, -7, 1, 1, 1, 0, -15, 6, 53, -23, -67, 22, 38, -8, -10, 1, 1, 0, -3, 3, 55, -28, -189, 81, 261, -90, -182, 46, 68, -11, -13, 1, 1, -1, 0, 30, -12, -229, 106, 691, -292, -1010, 359, 817, -229, -387, 79, 107, -14, -16, 1, 1

Offset: 0

Stephen O'Sullivan, Jan 10 2016

The entry a(n,k), n >= 0, k = 0,1,...,g, where g=3n, of this irregular triangle is the coefficient of (1+q^2)^k*q^(g-k) in the representation of the Gaussian polynomial [2n+3,3]q = Sum{k=0..g) a(n,k)*(1+q^2)^k*q^(g-k).
Row n is of length 3n+1.
The sequence arises in the formal derivation of the stability polynomial B(x) = Sum_{i=0..N} d_i T(iM,x) of rank N, and degree L, where T(iM,x) denotes the Chebyshev polynomial of the first kind of degree iM (A053120). The coefficients d_i are determined by order conditions on the stability polynomial.
Conjecture: More generally, the Gaussian polynomial [2*n+m+1-(m mod 2),m]q = Sum{k=0..g(m;n)} a(m;n,k)*(1+q^2)^k*q^(g(m;n)-k), for m >= 0, n >= 0, where g(m;n) = m*n if m is odd and (2*n+1)*m/2 if m is even, and the tabf array entries a(m;n,k) are the coefficients of the g.f. for the row n polynomials G(m;n,x) = (d^m/dt^m)G(m;n,t,x)/m!|{t=0}, with G(m;n,t,x) = (1+t)*Product{k=1..n+(m - m (mod 2))/2}(1 + t^2 + 2*t*T(k,x/2) (Chebyshev's T-polynomials). Hence a(m;n,k) = [x^k]G(m;n,x), for k=0..g(m;n). The present entry is the instance m = 3. (Thanks to Wolfdieter Lang for clarifying the text on the general prescription of a(m;n,k).)
From Robert Israel, Jan 15 2016: (Start)
a(n,0) = A056594(n).
a(n,1) = (-1)^((n+1)/2) * A142150(n+1).
a(2n,2) = 5*(-1)^(n+1)*A000217(n), a(2n+1,2) =(-1)^n*(n+1).
It appears that Sum_{j=0..k+1} C(k+1,j)*a(n+2*j,k) = 0.
(End)

			The irregular triangle a(n, k) begins:
n/k 0  1   2   3   4    5   6   7   8   9  10 11 12
0:  1
1:  0 -1   1   1
2: -1  0   5  -2  -4   1    1
3:  0  2  -2 -15   7   17  -5  -7   1   1
4:  1  0 -15   6  53  -23 -67  22  38  -8 -10  1  1
...
Row n=5: 0 -3   3  55 -28 -189  81 261 -90 -182  46 68 -11 -13 1 1;
Row n=6: -1  0 30 -12 -229  106 691 -292 -1010 359 817 -229 -387 79 107 -14 -16 1 1.
Row n=7: 0 4 -4 -134 70 896 -416 -2561 1073 3903 -1415 -3529 1057 1991 -467 -709 121 155 -17 -19 1 1.
... Reformatted and extended. - _Wolfdieter Lang_, Feb 13 2016

Stephen O'Sullivan, Table of n, a(n) for n = 0..1425
S. O'Sullivan, A class of high-order Runge-Kutta-Chebyshev stability polynomials, Journal of Computational Physics, 300 (2015), 665-678.
Wikipedia, Gaussian binomial coefficients.

Cf. A000217. A056594, A142150, A267482, A267483, A267484, A267485, A267486.

A267120 := proc (n, k) local y: y := expand(subs(t = 0, diff((1+t)*product(1+t^2+2*t*ChebyshevT(i, x/2), i = 1 .. n+1),t$3)/3!)): if k = 0 then subs(x = 0, y) else subs(x = 0, diff(y, x$k)/k!) end if: end proc: seq(seq(A267120(n, k), k = 0 .. 3*n), n = 0 .. 20);
# More efficient:
N:= 20: # to get rows 0 to N
P[0]:= (1+t)*(t^2 + t*x + 1):
B[0]:= 1:
for n from 1 to N do
  P[n]:= expand(series(P[n-1]*(1+t^2+2*t*orthopoly[T](n+1,x/2)),t,4));
  B[n]:= coeff(P[n],t,3);
od:
seq(seq(coeff(B[n],x,j),j=0..3*n),n=0..N); # Robert Israel, Jan 15 2016

row[n_] := 1/3! D[(1+t)*Product[1+t^2+2*t*ChebyshevT[i, x/2], {i, 1, n+1}], {t, 3}] /. t -> 0 // CoefficientList[#, x]&; Table[row[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Jan 16 2016 *)

G.f. for row polynomial: G(n,x) = (d^3/dt^3)((1+t)*Product_{i=1..n+1}(1+t^2+2t*T(i,x/2))/3!)|_{t=0}.

A267483 Triangle of coefficients of Gaussian polynomials [2n+3,2]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=2n+1.

Original entry on oeis.org

1, 1, 0, -1, 1, 1, 1, 2, -2, -3, 1, 1, 0, -2, 4, 7, -4, -5, 1, 1, 1, 3, -6, -13, 11, 16, -6, -7, 1, 1, 0, -3, 9, 22, -24, -40, 22, 29, -8, -9, 1, 1, 1, 4, -12, -34, 46, 86, -62, -91, 37, 46, -10, -11, 1, 1, 0, -4, 16, 50, -80, -166, 148, 239, -128, -174, 56, 67, -12, -13, 1, 1, 1, 5, -20, -70, 130, 296, -314, -553, 367, 541, -230, -297, 79, 92, -14, -15, 1, 1

Offset: 0

Stephen O'Sullivan, Jan 15 2016

The entry a(n,k), n >= 0, k = 0,1,...,g, where g=2n+1, of this irregular triangle is the coefficient of (1+q^2)^k*q^(g-k) in the representation of the Gaussian polynomial [2n+3,2]q = Sum{k=0..g) a(n,k)*(1+q^2)^k*q^(g-k).
Row n is of length 2n+2.
The sequence arises in the formal derivation of the stability polynomial B(x) = sum_{i=0..N} d_i T(iM,x) of rank N, and degree L, where T(iM,x) denotes the Chebyshev polynomial of the first kind (A053120) of degree iM. The coefficients d_i are determined by order conditions on the stability polynomial.
Conjecture: More generally, the Gaussian polynomial [2*n+m+1-(m mod 2),m]q = Sum{k=0..g(m;n)} a(m;n,k)*(1+q^2)^k*q^(g(m;n)-k), for m >= 0, n >= 0, where g(m;n) = m*n if m is odd and (2*n+1)*m/2 if m is even, and the tabf array entries a(m;n,k) are the coefficients of the g.f. for the row n polynomials G(m;n,x) = (d^m/dt^m)G(m;n,t,x)/m!|{t=0}, with G(m;n,t,x) = (1+t)*Product{k=1..n+(m - m (mod 2))/2}(1 + t^2 + 2*t*T(k,x/2) (Chebyshev's T-polynomials). Hence a(m;n,k) = [x^k]G(m;n,x), for k=0..g(m;n). The present entry is the instance m = 2. (Thanks to Wolfdieter Lang for clarifying the text on the general prescription of a(m;n,k).)

			1,1;
0,-1,1,1;
1,2,-2,-3,1,1;
0,-2,4,7,-4,-5,1,1;
1,3,-6,-13,11,16,-6,-7,1,1;
0,-3,9,22,-24,-40,22,29,-8,-9,1,1;
1,4,-12,-34,46,86,-62,-91,37,46,-10,-11,1,1;
0,-4,16,50,-80,-166,148,239,-128,-174,56,67,-12,-13,1,1;
1,5,-20,-70,130,296,-314,-553,367,541,-230,-297,79,92,-14,-15,1,1;

Stephen O'Sullivan, Table of n, a(n) for n = 0..991
S. O'Sullivan, A class of high-order Runge-Kutta-Chebyshev stability polynomials, Journal of Computational Physics, 300 (2015), 665-678.
Wikipedia, Gaussian binomial coefficients.

Cf. A053120, A267120, A267482, A267484, A267485, A267486.

A267483 := proc (n, k) local y: y := expand(subs(t = 0, diff((1+t)*product(1+t^2+2*t*ChebyshevT(i, x/2), i = 1 .. n+1),t$2)/2)): if k = 0 then subs(x = 0, y) else subs(x = 0, diff(y, x$k)/k!) end if: end proc: seq(seq(A267483(n, k), k = 0 .. 2*n+1), n = 0 .. 20);
# More efficient:
N:= 20: # to get rows 0 to N
P[0]:= (1+t)*(t^2 + t*x + 1):
B[0]:= 1+x:
for n from 1 to N do
  P[n]:= expand(series(P[n-1]*(1+t^2+2*t*orthopoly[T](n+1,x/2)),t,3));
  B[n]:= coeff(P[n],t,2);
od:
seq(seq(coeff(B[n],x,j),j=0..2*n+1),n=0..N); # From A267120 entry by Robert Israel

row[n_] := 1/2! D[(1+t)*Product[1+t^2+2*t*ChebyshevT[i, x/2], {i, 1, n+1}], {t, 2}] /. t -> 0 // CoefficientList[#, x]&; Table[row[n], {n, 0, 20}] // Flatten (* From A267120 entry by Jean-François Alcover *)

G.f. for row polynomial: G(n,x) = (d^2/dt^2)((1+t)*Product_{i=1..n+1}(1+t^2+2t*T(i,x/2)))/2!|_{t=0}.

A267484 Triangle of coefficients of Gaussian polynomials [2n+5,4]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=4n+2.

Original entry on oeis.org

-1, 1, 1, -1, 0, 5, -2, -4, 1, 1, -2, 2, 17, -9, -32, 12, 24, -6, -8, 1, 1, -2, 0, 31, -12, -121, 52, 187, -67, -143, 38, 58, -10, -12, 1, 1, -3, 3, 64, -37, -357, 168, 883, -361, -1154, 397, 875, -239, -399, 80, 108, -14, -16, 1, 1, -3, 0, 94, -36, -808, 366, 3019, -1312, -6023, 2351, 7182, -2415, -5439, 1512, 2686, -587, -863, 138, 174, -18, -20, 1, 1, -4, 4, 158, -94, -1720, 856, 8611, -3923, -23883, 10003, 40648, -15328, -45241, 14957, 34203, -9623, -17893, 4135, 6485, -1175, -1599, 212, 256, -22, -24, 1, 1

Offset: 0

Stephen O'Sullivan, Jan 15 2016

The entry a(n,k), n >= 0, k = 0,1,...,g, where g=4n+2, of this irregular triangle is the coefficient of (1+q^2)^k*q^(g-k) in the representation of the Gaussian polynomial [2*n+5,4]q = Sum{k=0..g) a(n,k)*(1+q^2)^k*q^(g-k).
Row n is of length 4n+3.
The sequence arises in the formal derivation of the stability polynomial B(x) = Sum_{i=0..N} d_i T(iM,x) of rank N, and degree L, where T(iM,x) denotes the Chebyshev polynomial of the first kind of degree iM. The coefficients d_i are determined by order conditions on the stability polynomial.
Conjecture: More generally, the Gaussian polynomial [2*n+m+1-(m mod 2),m]q = Sum{k=0..g(m;n)} a(m;n,k)*(1+q^2)^k*q^(g(m;n)-k), for m >= 0, n >= 0, where g(m;n) = m*n if m is odd and (2*n+1)*m/2 if m is even, and the tabf array entries a(m;n,k) are the coefficients of the g.f. for the row n polynomials G(m;n,x) = (d^m/dt^m)G(m;n,t,x)/m!|{t=0}, with G(m;n,t,x) = (1+t)*Product{k=1..n+(m - m (mod 2))/2}(1 + t^2 + 2*t*T(k,x/2) (Chebyshev's T-polynomials). Hence a(m;n,k) = [x^k]G(m;n,x), for k=0..g(m;n). The present entry is the instance m = 2. (Thanks to Wolfdieter Lang for clarifying the text on the general prescription of a(m;n,k).)

			-1,1,1;
-1,0,5,-2,-4,1,1;
-2,2,17,-9,-32,12,24,-6,-8,1,1;
-2,0,31,-12,-121,52,187,-67,-143,38,58,-10,-12,1,1;
-3,3,64,-37,-357,168,883,-361,-1154,397,875,-239,-399,80,108,-14,-16,1,1;

Stephen O'Sullivan, Table of n, a(n) for n = 0..902
S. O'Sullivan, A class of high-order Runge-Kutta-Chebyshev stability polynomials, Journal of Computational Physics, 300 (2015), 665-678.
Wikipedia, Gaussian binomial coefficients.

Cf. A267120, A267482, A267483, A267485, A267486.

A267484 := proc (n, k) local y: y := expand(subs(t = 0, diff((1+t)*product(1+t^2+2*t*ChebyshevT(i, x/2), i = 1 .. n+2),t$4)/4!)): if k = 0 then subs(x = 0, y) else subs(x = 0, diff(y, x$k)/k!) end if: end proc: seq(seq(A267484(n, k), k = 0 .. 4*n+2), n = 0 .. 20);

row[n_] := 1/4! D[(1+t)*Product[1+t^2+2*t*ChebyshevT[i, x/2], {i, 1, n+1}], {t, 4}] /. t -> 0 // CoefficientList[#, x]&; Table[row[n], {n, 0, 20}] // Flatten (* From A267120 entry by Jean-François Alcover *)

G.f. for row polynomial: G(n,x) = (d^4/dt^4)((1+t)*Product_{i=1..n+1}(1+t^2+2t*T(i,x/2))/4!)|_{t=0}.

A267485 Triangle of coefficients of Gaussian polynomials [2n+5,5]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=5n.

Original entry on oeis.org

1, 1, 2, -2, -3, 1, 1, -2, 2, 17, -9, -32, 12, 24, -6, -8, 1, 1, -2, -6, 25, 71, -80, -218, 126, 284, -106, -190, 48, 69, -11, -13, 1, 1, 3, -6, -70, 101, 506, -453, -1592, 980, 2658, -1201, -2608, 886, 1581, -400, -600, 108, 139, -16, -18, 1, 1, 3, 12, -88, -334, 779, 2774, -3226, -10389, 7709, 21620, -11608, -27865, 11496, 23591, -7645, -13512, 3427, 5276, -1020, -1385, 193, 234, -21, -23, 1, 1

Offset: 0

Stephen O'Sullivan, Jan 15 2016

The entry a(n,k), n >= 0, k = 0,1,...,g, where g=5n, of this irregular triangle is the coefficient of (1+q^2)^k*q^(g-k) in the representation of the Gaussian polynomial [2n+5,5]q = Sum{k=0..g) a(n,k)*(1+q^2)^k*q^(g-k).
Row n is of length 5n+1.
The sequence arises in the formal derivation of the stability polynomial B(x) = Sum_{i=0..N} d_i T(iM,x) of rank N, and degree L, where T(iM,x) denotes the Chebyshev polynomial of the first kind of degree iM. The coefficients d_i are determined by order conditions on the stability polynomial.
Conjecture: More generally, the Gaussian polynomial [2*n+m+1-(m mod 2),m]q = Sum{k=0..g(m;n)} a(m;n,k)*(1+q^2)^k*q^(g(m;n)-k), for m >= 0, n >= 0, where g(m;n) = m*n if m is odd and (2*n+1)*m/2 if m is even, and the tabf array entries a(m;n,k) are the coefficients of the g.f. for the row n polynomials G(m;n,x) = (d^m/dt^m)G(m;n,t,x)/m!|{t=0}, with G(m;n,t,x) = (1+t)*Product{k=1..n+(m - m (mod 2))/2}(1 + t^2 + 2*t*T(k,x/2) (Chebyshev's T-polynomials). Hence a(m;n,k) = [x^k]G(m;n,x), for k=0..g(m;n). The present entry is the instance m = 2. (Thanks to Wolfdieter Lang for clarifying the text on the general prescription of a(m;n,k).)

			1;
1,2,-2,-3,1,1;
-2,2,17,-9,-32,12,24,-6,-8,1,1;
-2,-6,25,71,-80,-218,126,284,-106,-190,48,69,-11,-13,1,1;

Stephen O'Sullivan, Table of n, a(n) for n = 0..1070
S. O'Sullivan, A class of high-order Runge-Kutta-Chebyshev stability polynomials, Journal of Computational Physics, 300 (2015), 665-678.
Wikipedia, Gaussian binomial coefficients.

Cf. A267120, A267482, A267483, A267484, A267486.

A267485 := proc (n, k) local y: y := expand(subs(t = 0, diff((1+t)*product(1+t^2+2*t*ChebyshevT(i, x/2), i = 1 .. n+2),t$5)/5!)): if k = 0 then subs(x = 0, y) else subs(x = 0, diff(y, x$k)/k!) end if: end proc: seq(seq(A267485(n, k), k = 0 .. 5*n), n = 0 .. 5);

row[n_] := 1/5! D[(1+t)*Product[1+t^2+2*t*ChebyshevT[i, x/2], {i, 1, n+1}], {t, 5}] /. t -> 0 // CoefficientList[#, x]&; Table[row[n], {n, 0, 20}] // Flatten (* From A267120 entry by Jean-François Alcover *)

G.f. for row polynomial: G(n,x) = (d^5/dt^5)((1+t)*Product_{i=1..n+1}(1+t^2+2t*T(i,x/2))/5!)|_{t=0}.

Added row length

A267486 Triangle of coefficients of Gaussian polynomials [2n+7,6]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=6n+3.

Original entry on oeis.org

-1, -2, 1, 1, 0, 2, -2, -15, 7, 17, -5, -7, 1, 1, -2, -6, 25, 71, -80, -218, 126, 284, -106, -190, 48, 69, -11, -13, 1, 1, 0, 6, -12, -137, 196, 945, -811, -2745, 1602, 4163, -1780, -3711, 1193, 2059, -493, -722, 123, 156, -17, -19, 1, 1, -3, -12, 94, 358, -952, -3430, 4699, 15615, -13467, -39946, 24494, 63168, -29535, -65638, 24206, 46512, -13652, -22891, 5294, 7834, -1386, -1831, 234, 279, -23, -25, 1, 1

Offset: 0

Stephen O'Sullivan, Jan 15 2016

The entry a(n,k), n >= 0, k = 0,1,...,g, where g=6n+3, of this irregular triangle is the coefficient of (1+q^2)^k*q^(g-k) in the representation of the Gaussian polynomial [2n+7,6]q = Sum{k=0..g) a(n,k)*(1+q^2)^k*q^(g-k).
Row n is of length 6n+4.
The sequence arises in the formal derivation of the stability polynomial B(x) = Sum_{i=0..N} d_i T(iM,x) of rank N, and degree L, where T(iM,x) denotes the Chebyshev polynomial of the first kind of degree iM. The coefficients d_i are determined by order conditions on the stability polynomial.
Conjecture: More generally, the Gaussian polynomial [2*n+m+1-(m mod 2),m]q = Sum{k=0..g(m;n)} a(m;n,k)*(1+q^2)^k*q^(g(m;n)-k), for m >= 0, n >= 0, where g(m;n) = m*n if m is odd and (2*n+1)*m/2 if m is even, and the tabf array entries a(m;n,k) are the coefficients of the g.f. for the row n polynomials G(m;n,x) = (d^m/dt^m)G(m;n,t,x)/m!|{t=0}, with G(m;n,t,x) = (1+t)*Product{k=1..n+(m - m (mod 2))/2}(1 + t^2 + 2*t*T(k,x/2) (Chebyshev's T-polynomials). Hence a(m;n,k) = [x^k]G(m;n,x), for k=0..g(m;n). The present entry is the instance m = 2. (Thanks to Wolfdieter Lang for clarifying the text on the general prescription of a(m;n,k).)

			-1,-2,1,1;
0,2,-2,-15,7,17,-5,-7,1,1;
-2,-6,25,71,-80,-218,126,284,-106,-190,48,69,-11,-13,1,1;

Stephen O'Sullivan, Table of n, a(n) for n = 0..1343
S. O'Sullivan, A class of high-order Runge-Kutta-Chebyshev stability polynomials, Journal of Computational Physics, 300 (2015), 665-678.
Wikipedia, Gaussian binomial coefficients.

Cf. A267120, A267482, A267483, A267484, A267485.

A267486 := proc (n, k) local y: y := expand(subs(t = 0, diff((1+t)*product(1+t^2+2*t*ChebyshevT(i, x/2), i = 1 .. n+3),t$6)/6!)): if k = 0 then subs(x = 0, y) else subs(x = 0, diff(y, x$k)/k!) end if: end proc: seq(seq(A267486(n, k), k = 0 .. 6*n+3), n = 0 .. 20);

row[n_] := 1/6! D[(1+t)*Product[1+t^2+2*t*ChebyshevT[i, x/2], {i, 1, n+1}], {t, 6}] /. t -> 0 // CoefficientList[#, x]&; Table[row[n], {n, 0, 20}] // Flatten (* From A267120 entry by Jean-François Alcover *)

G.f. for row polynomial: G(n,x) = (d^6/dt^6)((1+t)*Product_{i=1..n+1}(1+t^2+2t*T(i,x/2))/6!)|_{t=0}.

Added row length

cp's OEIS Frontend

A187660 Triangle read by rows: T(n,k) = (-1)^(floor(3*k/2))*binomial(floor((n+k)/2),k), 0 <= k <= n.

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A267120 Triangle of coefficients of Gaussian polynomials [2n+3,3]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=3n.

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A267483 Triangle of coefficients of Gaussian polynomials [2n+3,2]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=2n+1.

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A267484 Triangle of coefficients of Gaussian polynomials [2n+5,4]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=4n+2.

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A267485 Triangle of coefficients of Gaussian polynomials [2n+5,5]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=5n.

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A267486 Triangle of coefficients of Gaussian polynomials [2n+7,6]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=6n+3.

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A187660 Triangle read by rows: T(n,k) = (-1)^(floor(3k/2))binomial(floor((n+k)/2),k), 0 <= k <= n.