cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Stephen O'Sullivan, Jan 15 2016

Keywords

Comments

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

Examples

			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;

Links

Crossrefs

Cf. A049310, A267120, A267483, A267484, A267485, A267486, A294099.
Cf. A046854, A066170, A130777, A187660 all signed versions.

Programs

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

Formula

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

Views

Author

Stephen O'Sullivan, Jan 15 2016

Keywords

Comments

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).)

Examples

			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;

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

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

Views

Author

Stephen O'Sullivan, Jan 15 2016

Keywords

Comments

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).)

Examples

			-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;

Links

Crossrefs

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

Programs

  • Maple
    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);
  • Mathematica
    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 *)

Formula

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

Views

Author

Stephen O'Sullivan, Jan 15 2016

Keywords

Comments

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).)

Examples

			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;

Links

Crossrefs

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

Programs

  • Maple
    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);
  • Mathematica
    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 *)

Formula

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}.

Extensions

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

Views

Author

Stephen O'Sullivan, Jan 15 2016

Keywords

Comments

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).)

Examples

			-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;

Links

Crossrefs

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

Programs

  • Maple
    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);
  • Mathematica
    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 *)

Formula

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}.

Extensions

Added row length
Showing 1-5 of 5 results.

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