Stephen O'Sullivan - cp's OEIS frontend

User: Stephen O'Sullivan

Stephen O'Sullivan has authored 6 sequences.

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.

-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

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

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}.

A267482 Triangle of coefficients of Gaussian polynomials [2n+1,1]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=n.

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

Offset: 0

Stephen O'Sullivan, Jan 15 2016

The entry a(n,k), n >= 0, k = 0,1,...,g, where g=n, of this irregular triangle is the coefficient of (1+q^2)^k*q^(g-k) in the representation of the Gaussian polynomial [2n+1,1]q = Sum{k=0..g) a(n,k)*(1+q^2)^k*q^(g-k).
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).)
Signed version of A046854, A130777.
Conjecture: row n is U(n, x/2) + U(n-1, x/2) where U is the sequence of Chebyshev polynomials of the second kind. - Thomas Baruchel, Jun 03 2018 [For a proof see the following comment.]
From Wolfdieter Lang, Oct 19 2019: (Start)
The row polynomial R(n, x) = Sum_{k=0..n} a(n, k)*x^k = [2*n+1]_q / q^n with the q-number [2*n+1]_q := (1 - q^n)/(1 - q), which for q = 1 becomes 2*n+1, and x = x(q) = q + q^(-1). See the simplified Name and the first comment. In terms of Chebyshev S polynomials (A049310) this q-number is written as [2*n+1]_q = q^n*S(2*n, q^(1/2) + q^(-1/2)), hence R(n, x) = S(2*n, sqrt(2+x)) = S(n, x) + S(n-1, x) (which proves the conjecture of the previous comment).
For the o.g.f. of R(n, x) see the formula section.
My motivation for looking at this sequence came from the Brändli and Beyne paper's recurrence for the polynomial P_m(s) which coincides with R(n, x), with m -> n and s -> x. (End)
A294099(n, k) = Sum_{j=0..k} n^j * T(n, j) for all n, k in Z. - Michael Somos, Jun 19 2023

			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;

Stephen O'Sullivan, Table of n, a(n) for n = 0..495
Gerold Brändli and Tim Beyne, Modified Congruence Modulo n with Half the Amount of Residues, arXiv:1504.02757 [math.NT], 2016-2017. Definition 6.for polynomials P_m(s).
Stephen O'Sullivan, A class of high-order Runge-Kutta-Chebyshev stability polynomials, Journal of Computational Physics, 300 (2015), 665-678.
Wikipedia, Gaussian binomial coefficients.
Index entries for sequences related to Chebyshev polynomials.

Cf. A049310, A267120, A267483, A267484, A267485, A267486, A294099.

Cf. A046854, A066170, A130777, A187660 all signed versions.

A267482 := 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),t))): if k = 0 then subs(x = 0, y) else subs(x = 0, diff(y, x$k)/k!) end if: end proc: seq(seq(A267482(n, k), k = 0 .. n), n = 0 .. 20);

row[n_] := D[(1+t)*Product[1+t^2+2*t*ChebyshevT[i, x/2], {i, 1, n}], t] /. t -> 0 // CoefficientList[#, x]&; Table[row[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jan 16 2016 *)

T(n,k) = (-1)^((n-k)\2)*binomial((n+k)\2, k); \\François Marques, Sep 28 2021

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)))|_{t=0}.
From Wolfdieter Lang, Oct 19 2019: (Start)
Row polynomial R(n, x) = S(2*n, sqrt(2+x)) = S(n, x) + S(n-1, x) = Sum_{k=0..n} (-1)^k*binomial(2*n-k, k)*(2 + x)^(n-k), for n >= 0. (See the Thomas Baruchel conjecture and the proof above.) For the S(n, x) coefficients see A049310.
R(n, x) = Sum_{j=0} (-1)^e(n,j)*binomial(e(n,j) + j, j)*x^j*, with e(n,j) := floor((n-j)/2). See eq. (12) of the Brändli and Beyne paper.
G.f. for row polynomials R(n, x) (that is of the triangle): G(x,z) = (1 + z)/(1 - x*z + z^2).
Recurrence for R(n, x): R(-1, x)  = -1, R(0, x) = 1, R(n, x) = x*R(n-1, x) - R(n-2, x), for n >= 1. (See the Brändli and Beyne link, polynomials P_m(s) in Definition 6.)
(End)
T(n,k) = (-1)^(floor((n-k)/2))*binomial(floor((n+k)/2), k). - François Marques, Sep 28 2021

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}.

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}.

cp's OEIS Frontend

User: Stephen O'Sullivan

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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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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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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A267482 Triangle of coefficients of Gaussian polynomials [2n+1,1]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=n.

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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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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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