A123583 -id:A123583 - cp's OEIS frontend

A136667 Triangle read by rows: T(n, k) is the coefficient of x^k in the polynomial 1 - T_n(x)^2, where T_n(x) is the n-th Hermite polynomial of the Hochstadt kind (A137286) as related to the generalized Chebyshev in a Shabat way (A123583): p(x,n)=x*p(x,n-1)-p(x,n-2); q(x,n)=1-p(x,n)^2.

0, 1, 0, -1, -3, 0, 4, 0, -1, 1, 0, -25, 0, 10, 0, -1, -63, 0, 144, 0, -97, 0, 18, 0, -1, 1, 0, -1089, 0, 924, 0, -262, 0, 28, 0, -1, -2303, 0, 8352, 0, -9489, 0, 3576, 0, -574, 0, 40, 0, -1, 1, 0, -77841, 0, 103230, 0, -49291, 0, 10548, 0, -1099, 0, 54, 0, -1, -147455, 0, 748800, 0, -1215585, 0, 699630, 0, -188043, 0

Offset: 1

Roger L. Bagula, Apr 02 2008

Row sums are {0, 0, 0, -15, 1, -399, -399, -14399, -78399, -639999, -12959999}.

			The irregular triangle begins
  {0},
  {1, 0, -1},
  {-3, 0, 4, 0, -1},
  {1, 0, -25, 0, 10, 0, -1},
  {-63, 0, 144, 0, -97, 0, 18, 0, -1},
  {1, 0, -1089, 0, 924, 0, -262,0, 28, 0, -1},
  {-2303, 0, 8352, 0, -9489, 0, 3576, 0, -574, 0, 40, 0, -1},
  {1, 0, -77841, 0, 103230, 0, -49291, 0, 10548,0, -1099, 0, 54, 0, -1},
  ...

Defined: page 8 and pages 42 - 43: Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986
G. B. Shabat and I. A. Voevodskii, Drawing curves over number fields, The Grothendieck Festschift, vol. 3, Birkhäuser, 1990, pp. 199-22

G. B. Shabat and A. Zvonkin, Plane trees and algebraic numbers, Contemporary Math., 1994, vol. 178, 233-275.

Cf. A123583, A137286, A136247.

P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = x*P[x, n - 1] - n*P[x, n - 2]; Q[x_, n_] := Q[x, n] = 1 - P[x, n]^2; Table[ExpandAll[Q[x, n]], {n, 0, 10}]; a = Join[{{0}}, Table[CoefficientList[Q[x, n], x], {n, 0, 10}]]; Flatten[a]

polx(n) = if (n == 0, 1, if (n == 1, x, x*polx(n - 1) - n*polx(n - 2)));
tabf(nn) = {for (n = 0, nn, pol = 1 - polx(n)^2; for (i = 0, 2*n, print1(polcoeff(pol, i), ", "); ); print(); ); }  \\ Michel Marcus, Feb 26 2018

out=1-A137286(x,n)^2; p(x,n)=x*p(x,n-1)-p(x,n-2); q(x,n)=1-p(x,n)^2.

Keyword changed to tabf by Michel Marcus, Feb 26 2018

A156647 Square array T(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2) with T(n, 0) = n!, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, -3, 2, 1, -8, 144, 6, 1, -15, 2304, -97200, 24, 1, -24, 14400, -22579200, 914457600, 120, 1, -35, 57600, -857304000, 7517247897600, -119833267276800, 720, 1, -48, 176400, -13548902400, 3163657512960000, -85018329720343756800, 218719679433615360000, 5040

Offset: 0

Roger L. Bagula, Feb 12 2009

			Square array begins as:
    1,                1,                     1, ...;
    1,               -3,                    -8, ...;
    2,              144,                  2304, ...;
    6,           -97200,             -22579200, ...;
   24,        914457600,         7517247897600, ...;
  120, -119833267276800, -85018329720343756800, ...;
Triangle begins as:
  1;
  1,   1;
  1,  -3,     2;
  1,  -8,   144,          6;
  1, -15,  2304,     -97200,            24;
  1, -24, 14400,  -22579200,     914457600,              120;
  1, -35, 57600, -857304000, 7517247897600, -119833267276800, 720;

G. C. Greubel, Antidiagonal rows n = 0..25, flattened

Cf. A123583.

T:= func< n,k | n eq 0 select 1 else k eq 0 select Factorial(n) else (&*[1 - Evaluate(ChebyshevT(j), k+1)^2 : j in [1..n]]) >;
[T(k, n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 02 2021

T[n_, k_]= If[k==0, n!, Product[1 - ChebyshevT[j, k+1]^2, {j,n}]];
Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jul 02 2021 *)

def T(n,k): return factorial(n) if (k==0) else product( 1 - chebyshev_T(j, k+1)^2 for j in (1..n) )
flatten([[T(k, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 02 2021

T(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2) with T(n, 0) = n! (square array).

Edited by G. C. Greubel, Jul 02 2021

A156645 Triangle T(n, k, m) = b(n,m)/(b(k,m)*b(n-k,m)), where b(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2), b(n, 0) = n!, and m = 2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 36, 1, 1, 1225, 1225, 1, 1, 41616, 1416100, 41616, 1, 1, 1413721, 1634261476, 1634261476, 1413721, 1, 1, 48024900, 1885939157025, 64069586905104, 1885939157025, 48024900, 1, 1, 1631432881, 2176372249076025, 2511659716192658889, 2511659716192658889, 2176372249076025, 1631432881, 1

Offset: 0

Roger L. Bagula, Feb 12 2009

			Triangle begins as:
  1;
  1,        1;
  1,       36,             1;
  1,     1225,          1225,              1;
  1,    41616,       1416100,          41616,             1;
  1,  1413721,    1634261476,     1634261476,       1413721,        1;
  1, 48024900, 1885939157025, 64069586905104, 1885939157025, 48024900, 1;

G. C. Greubel, Rows n = 0..25 of the triangle, flattened

Cf. A007318 (m=0), A173585 (m=1), this sequence (m=2), A156646 (m=10).

Cf. A123583, A156647.

b:= func< n, k | n eq 0 select 1 else k eq 0 select Factorial(n) else (&*[1 - Evaluate(ChebyshevT(j), k+1)^2 : j in [1..n]]) >;
T:= func< n,k,m | b(n,m)/(b(k,m)*b(n-k,m)) >;
[T(n,k,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 03 2021

(* First program *)
b[n_, k_]:= b[n,k]= If[k==0, n!, Product[1 -ChebyshevT[j, k+1]^2, {j,n}]];
T[n_, k_, m_]= b[n,m]/(b[k,m]*b[n-k,m]);
Table[T[n, k, 2], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jul 03 2021 *)
(* Second program *)
T[n_, k_, m_]:= T[n, k, m]= Product[(1 - ChebyshevT[2*j+2*k, m+1])/(1 - ChebyshevT[2*j, m+1]), {j, n-k}];
Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 03 2021 *)

def b(n, k): return factorial(n) if (k==0) else product( 1 - chebyshev_T(j, k+1)^2 for j in (1..n) )
def T(n, k, m): return b(n,m)/(b(k,m)*b(n-k,m))
flatten([[T(n, k, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 03 2021

T(n, k, m) = b(n,m)/(b(k,m)*b(n-k,m)), where b(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2), b(n, 0) = n!, and m = 2.
From G. C. Greubel, Jul 03 2021: (Start)
T(n, k, m) = b(n,m)/(b(k,m)*b(n-k,m)), where b(n, k) = (1/2^n)*Product_{j=1..n} (1 - ChebyshevT(2*j, k+1)), b(n, 0) = n!, and m = 2.
T(n, k, m) = Product_{j=1..n-k} ( (1 - ChebyshevT(2*j+2*k, m+1))/(1 - ChebyshevT(2*j, m+1)) ) with m = 2. (End)

Edited by G. C. Greubel, Jul 03 2021

A156646 Triangle T(n, k, m) = b(n,m)/(b(k,m)*b(n-k,m)), where b(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2), b(n, 0) = n!, and m = 10, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 484, 1, 1, 233289, 233289, 1, 1, 112444816, 54198633636, 112444816, 1, 1, 54198168025, 12591535188240100, 12591535188240100, 54198168025, 1, 1, 26123404543236, 2925290638056514680225, 1409984043580226203632400, 2925290638056514680225, 26123404543236, 1

Offset: 0

Roger L. Bagula, Feb 12 2009

			Triangle begins as:
  1;
  1,           1;
  1,         484,                 1;
  1,      233289,            233289,                 1;
  1,   112444816,       54198633636,         112444816,           1;
  1, 54198168025, 12591535188240100, 12591535188240100, 54198168025, 1;

G. C. Greubel, Rows n = 0..25 of the triangle, flattened

Cf. A007318 (m=0), A173585 (m=1), A156645 (m=2), this sequence (m=10).

Cf. A123583, A156647.

b:= func< n, k | n eq 0 select 1 else k eq 0 select Factorial(n) else (&*[1 - Evaluate(ChebyshevT(j), k+1)^2 : j in [1..n]]) >;
T:= func< n,k,m | b(n,m)/(b(k,m)*b(n-k,m)) >;
[T(n,k,10): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 03 2021

(* First program *)
b[n_, k_]:= b[n,k]= If[k==0, n!, Product[1 -ChebyshevT[j, k+1]^2, {j,n}]];
T[n_, k_, m_]= b[n,m]/(b[k,m]*b[n-k,m]);
Table[T[n,k,10], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jul 03 2021 *)
(* Second program *)
T[n_, k_, m_]:= T[n, k, m]= Product[(1 - ChebyshevT[2*j+2*k, m+1])/(1 - ChebyshevT[2*j, m+1]), {j, n-k}];
Table[T[n,k,12], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 03 2021 *)

def b(n, k): return factorial(n) if (k==0) else product( 1 - chebyshev_T(j, k+1)^2 for j in (1..n) )
def T(n, k, m): return b(n,m)/(b(k,m)*b(n-k,m))
flatten([[T(n, k, 10) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 03 2021

T(n, k, m) = b(n,m)/(b(k,m)*b(n-k,m)), where b(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2), b(n, 0) = n!, and m = 10.
From G. C. Greubel, Jul 03 2021: (Start)
T(n, k, m) = b(n,m)/(b(k,m)*b(n-k,m)), where b(n, k) = (1/2^n)*Product_{j=1..n} (1 - ChebyshevT(2*j, k+1)), b(n, 0) = n!, and m = 10.
T(n, k, m) = Product_{j=1..n-k} ( (1 - ChebyshevT(2*j+2*k, m+1))/(1 - ChebyshevT(2*j, m+1)) ) with m = 10. (End)

Edited by G. C. Greubel, Jul 03 2021

A173585 Triangle T(n, k, q) = c(n, q)/(c(k, q)c(n-k, q)), where c(n, q) = Product_{j=1..n} t(2j, q), t(n, q) = (1/4)*( (2 + sqrt(q))^n + (2 - sqrt(q))^n - 2 ), and q = 3, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 16, 1, 1, 225, 225, 1, 1, 3136, 44100, 3136, 1, 1, 43681, 8561476, 8561476, 43681, 1, 1, 608400, 1660970025, 23150231104, 1660970025, 608400, 1, 1, 8473921, 322220846025, 62555239000969, 62555239000969, 322220846025, 8473921, 1

Offset: 0

Roger L. Bagula, Feb 22 2010

			Triangle begins as:
  1;
  1,      1;
  1,     16,          1;
  1,    225,        225,           1;
  1,   3136,      44100,        3136,          1;
  1,  43681,    8561476,     8561476,      43681,      1;
  1, 608400, 1660970025, 23150231104, 1660970025, 608400, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A022168 (q=0), A022173 (q=1), this sequence (q=3).
Cf. A007318 (m=0), this sequence (m=1), A156645 (m=2), A156646 (m=10).
Cf. A123583, A156647.

b:= func< n, k | n eq 0 select 1 else k eq 0 select Factorial(n) else (&*[1 - Evaluate(ChebyshevT(j), k+1)^2 : j in [1..n]]) >;
T:= func< n,k,m | b(n,m)/(b(k,m)*b(n-k,m)) >;
[T(n,k,1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 06 2021

(* First program *)
f[n_, q_]:= (1/4)*((2+Sqrt[q])^n + (2-Sqrt[q])^n -2);
c[n_, q_]:= Product[f[k, q], {k, 2, n, 2}]//Simplify;
T[n_, k_, q_]:= c[n, q]/(c[k, q]*c[n - k, q]);
Table[T[n, k, 3], {n, 0, 10, 2}, {k, 0, n, 2}]//Flatten (* modified by G. C. Greubel, Jul 06 2021 *)
(* Second program *)
t[n_, q_]:= (1/4)*(Round[(2+Sqrt[q])^n + (2-Sqrt[q])^n] -2);
c[n_, q_]:= Product[t[2*j, q], {j,n}];
T[n_, k_, q_]:= c[n, q]/(c[k, q]*c[n-k, q]);
Table[T[n, k, 3], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 06 2021 *)

@CachedFunction
def f(n,q): return (1/4)*( round((2 + sqrt(q))^n + (2 - sqrt(q))^n) - 2 )
def c(n,q): return product( f(2*j, q) for j in (1..n))
def T(n,k,q): return c(n, q)/(c(k, q)*c(n-k, q))
flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 06 2021

T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)), where c(n, q) = Product_{j=1..n} t(2*j, q), t(n, q) = (1/4)*( (2 + sqrt(q))^n + (2 - sqrt(q))^n - 2 ), and q = 3.
From G. C. Greubel, Jul 06 2021: (Start)
T(n, k, m) = b(n,m)/(b(k,m)*b(n-k,m)), where b(n, k) = (1/2^n)*Product_{j=1..n} (1 - ChebyshevT(2*j, k+1)), b(n, 0) = n!, and m = 1.
T(n, k, m) = Product_{j=1..n-k} ( (1 - ChebyshevT(2*j+2*k, m+1))/(1 - ChebyshevT(2*j, m+1)) ) with m = 1. (End)

Edited by G. C. Greubel, Jul 06 2021

A082649 Triangle of coefficients in expansion of sinh^2(n*x) in powers of sinh(x).

Original entry on oeis.org

1, 4, 4, 16, 24, 9, 64, 128, 80, 16, 256, 640, 560, 200, 25, 1024, 3072, 3456, 1792, 420, 36, 4096, 14336, 19712, 13440, 4704, 784, 49, 16384, 65536, 106496, 90112, 42240, 10752, 1344, 64, 65536, 294912, 552960, 559104, 329472, 114048, 22176, 2160, 81, 262144, 1310720, 2785280, 3276800, 2329600

Offset: 1

Gary W. Adamson, May 16 2003, suggested by Herb Conn

Using arcsin(x) = Pi/2 - arccos(x), valid for -1 < = x <= 1, we find sin^2(k*arcsin(x)) = sin^2(k*arccos(x)) for k odd, while sin^2(k*arcsin(x)) = 1 - sin^2(k*arccos(x)) for k even. Thus the expansion of sin^2(n*x) in powers of cos(x) will produce a similar table of coefficients. See the example section below. - Peter Bala, Feb 02 2017

			sinh^2 x = sinh^2 x
sinh^2 2x = 4 sinh^4 x + 4 sinh^2 x
sinh^2 3x = 16 sinh^6 x + 24 sinh^4 x + 9 sinh^2 x
sinh^2 4x = 64 sinh^8 x + 128 sinh^6 x + 80 sinh^4 x + 16 sinh^2 x
sinh^2 5x = 256 sinh^10 x + 640 sinh^8 x + 560 sinh^6 x + 200 sinh^4 x + 25 sinh^2 x
From _Peter Bala_, Feb 02 2016: (Start)
sin^2(x) = 1 - cos^2(x);
sin^2(2*x) = -4*cos^4(x) + 4*cos^2(x);
sin^2(3*x) = 1 - (16*cos^6(x) - 24*cos^4(x) + 9*cos^2(x));
sin^2(4*x) = -64*cos^8(x) + 128*cos^6(x) - 80*cos^4(x) + 16*cos^2(x);
sin^2(5*x) = 1 - (256*cos^10(x) - 640*cos^8(x) + 560*cos^6(x) - 200*cos^4(x) + 25*cos^2(x)). (End)

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 0 to 140, flattened)

A001108 gives row sums.

Closely related to A123583 and A123588.

g:= (1+x*y)/((1-x*y)*(1-(4+2*y)*x+x^2*y^2)):
S:= series(g,x,15):
seq(seq(coeff(coeff(S,x,n),y,k),k=0..n),n=0..14); # Robert Israel, Dec 20 2017

Table[Reverse[CoefficientList[1/x TrigExpand[Sinh[n ArcSinh[Sqrt[x]]]^2], x]], {n, 7}] // Flatten (* Eric W. Weisstein, Apr 05 2017 *)
Abs[Table[CoefficientList[x^n Piecewise[{{1 - ChebyshevT[n, 1/Sqrt[x]]^2, Mod[n, 2] == 0}, {ChebyshevT[n, 1/Sqrt[x]]^2, Mod[n, 2] == 1}}], x], {n, 10}]] // Flatten (* Eric W. Weisstein, Apr 05 2017 *)

Coefficients are: 4^(n-1), (2n)4^(n-2), (2n)(2n-3)4^(n-3)/2!, (2n)(2n-4)(2n-5)4^(n-4)/3!, (2n)(2n-5)(2n-6)(2n-7)4^(n-5)/4!, (2n)(2n-6)(2n-7)(2n-8)(2n-9)4^(n-6)/5!...

G.f. as triangle: (1+x*y)/((1-x*y)*(1-(4+2*y)*x+x^2*y^2)). - Robert Israel, Dec 20 2017

More terms from Robert Israel, Dec 20 2017

A138331 a(n) = C(n+5, 5)(n+3)(-1)^(n+1)*16/3.

Original entry on oeis.org

-16, 128, -560, 1792, -4704, 10752, -22176, 42240, -75504, 128128, -208208, 326144, -495040, 731136, -1054272, 1488384, -2062032, 2808960, -3768688, 4987136, -6517280, 8419840, -10764000, 13628160, -17100720, 21280896, -26279568, 32220160, -39239552

Offset: 0

Klaus Brockhaus, Mar 15 2008

Fourth column of the triangle defined in A123588, seventh column of the triangle defined in A123583.

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (-7,-21,-35,-35,-21,-7,-1).

Cf. A007318 (Pascal's triangle), A123588, A123583, A040977.

[ Binomial(n+5, 5)*(n+3)*(-1)^(n+1)*16/3: n in [0..28] ];

k:=3; [ Coefficients(1-ChebyshevT(n+k)^2)[2*k+1]: n in [0..28] ];

seq(binomial(n+5, 5)*(n+3)*(-1)^(n+1)*16/3, n=0..40); # Robert Israel, Oct 26 2017

LinearRecurrence[{-7,-21,-35,-35,-21,-7,-1},{-16,128,-560,1792,-4704,10752,-22176},30] (* Harvey P. Dale, May 27 2017 *)

for(n=0,28,print1(polcoeff(taylor(16*(x-1)/(x+1)^7,x),n),","));

a(n) = coefficient of x^6 in the polynomial 1 - T_(n+3)(x)^2, where T_n(x) is the n-th Chebyshev polynomial of the first kind.
G.f.: 16*(x-1)/(x+1)^7.
a(n) = (-1)^(n+1)*16*A040977(n).
a(n) = a(-n-5). - Bruno Berselli, Sep 13 2011

A138332 C(n+7, 7)(n+4)(-1)^(n+1)*16.

Original entry on oeis.org

-64, 640, -3456, 13440, -42240, 114048, -274560, 604032, -1235520, 2379520, -4356352, 7637760, -12899328, 21085440, -33488640, 51845376, -78450240, 116290944, -169206400, 242070400, -341003520, 473616000, -649284480, 879465600, -1178049600

Offset: 0

Klaus Brockhaus, Mar 15 2008

sign

Fifth column of the triangle defined in A123588, ninth column of the triangle defined in A123583.

Index entries for sequences related to Chebyshev polynomials.

Cf. A007318 (Pascal's triangle), A123588, A123583, A053347.

[ Binomial(n+7, 7)*(n+4)*(-1)^(n+1)*16: n in [0..24] ];

k:=4; [ Coefficients(1-ChebyshevT(n+k)^2)[2*k+1]: n in [0..24] ];

for(n=0,24,print1(polcoeff(taylor(64*(x-1)/(x+1)^9,x),n),","));

a(n) = coefficient of x^8 in the polynomial 1 - T_(n+4)(x)^2, where T_n(x) is the n-th Chebyshev polynomial of the first kind.
G.f.: 64*(x-1)/(x+1)^9.
a(n) = (-1)^(n+1)*64*A053347(n).

A138333 C(n+9, 9)(n+5)(-1)^(n+1)*256/5.

Original entry on oeis.org

-256, 3072, -19712, 90112, -329472, 1025024, -2818816, 7028736, -16180736, 34850816, -70946304, 137592832, -255836672, 458422272, -794962432, 1338884096, -2196606720, 3519493120, -5519205120, 8487198720, -12819206400, 19045678080, -27869287680

Offset: 0

Klaus Brockhaus, Mar 15 2008

sign

Sixth column of the triangle defined in A123588, eleventh column of the triangle defined in A123583.

Index entries for sequences related to Chebyshev polynomials.

Cf. A007318 (Pascal's triangle), A123588, A123583, A054334.

[ Binomial(n+9, 9)*(n+5)*(-1)^(n+1)*256/5: n in [0..22] ];

k:=5; [ Coefficients(1-ChebyshevT(n+k)^2)[2*k+1]: n in [0..22] ];

for(n=0,22,print1(polcoeff(taylor(256*(x-1)/(x+1)^11,x),n),","));

a(n) = coefficient of x^10 in the polynomial 1 - T_(n+5)(x)^2, where T_n(x) is the n-th Chebyshev polynomial of the first kind.
G.f.: 256*(x-1)/(x+1)^11.
a(n) = (-1)^(n+1)*256*A054334(n).

A138334 C(n+11, 11)(n+6)(-1)^(n+1)*512/3.

Original entry on oeis.org

-1024, 14336, -106496, 559104, -2329600, 8200192, -25346048, 70606848, -180590592, 429977600, -963149824, 2046693376, -4153583616, 8094162944, -15214592000, 27690557440, -48952949760, 84293314560, -141710499840, 233076480000

Offset: 0

Klaus Brockhaus, Mar 15 2008

sign

Seventh column of the triangle defined in A123588, thirteenth column of the triangle defined in A123583.

Index entries for sequences related to Chebyshev polynomials.

Cf. A007318 (Pascal's triangle), A123588, A123583.

[ Binomial(n+11, 11)*(n+6)*(-1)^(n+1)*512/3: n in [0..19] ];

k:=6; [ Coefficients(1-ChebyshevT(n+k)^2)[2*k+1]: n in [0..19] ];

Table[Binomial[n+11,11](n+6)(-1)^(n+1) 512/3,{n,0,20}] (* Harvey P. Dale, Jun 03 2021 *)

for(n=0,19,print1(polcoeff(taylor(1024*(x-1)/(x+1)^13,x),n),","));

a(n) = coefficient of x^12 in the polynomial 1 - T_(n+6)(x)^2, where T_n(x) is the n-th Chebyshev polynomial of the first kind.

G.f.: 1024*(x-1)/(x+1)^13.

cp's OEIS Frontend

A136667 Triangle read by rows: T(n, k) is the coefficient of x^k in the polynomial 1 - T_n(x)^2, where T_n(x) is the n-th Hermite polynomial of the Hochstadt kind (A137286) as related to the generalized Chebyshev in a Shabat way (A123583): p(x,n)=x*p(x,n-1)-p(x,n-2); q(x,n)=1-p(x,n)^2.

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A156647 Square array T(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2) with T(n, 0) = n!, read by antidiagonals.

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A156645 Triangle T(n, k, m) = b(n,m)/(b(k,m)*b(n-k,m)), where b(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2), b(n, 0) = n!, and m = 2, read by rows.

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A156646 Triangle T(n, k, m) = b(n,m)/(b(k,m)*b(n-k,m)), where b(n, k) = Product_{j=1..n} (1 - ChebyshevT(j, k+1)^2), b(n, 0) = n!, and m = 10, read by rows.

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A173585 Triangle T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)), where c(n, q) = Product_{j=1..n} t(2*j, q), t(n, q) = (1/4)*( (2 + sqrt(q))^n + (2 - sqrt(q))^n - 2 ), and q = 3, read by rows.

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A082649 Triangle of coefficients in expansion of sinh^2(n*x) in powers of sinh(x).

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A173585 Triangle T(n, k, q) = c(n, q)/(c(k, q)c(n-k, q)), where c(n, q) = Product_{j=1..n} t(2j, q), t(n, q) = (1/4)*( (2 + sqrt(q))^n + (2 - sqrt(q))^n - 2 ), and q = 3, read by rows.

A138331 a(n) = C(n+5, 5)(n+3)(-1)^(n+1)*16/3.

A138332 C(n+7, 7)(n+4)(-1)^(n+1)*16.

A138333 C(n+9, 9)(n+5)(-1)^(n+1)*256/5.

A138334 C(n+11, 11)(n+6)(-1)^(n+1)*512/3.