A123027 Triangle of coefficients of (1 - x)^n*U(n,-(3*x - 2)/(2*x - 2)), where U(n,x) is the n-th Chebyshev polynomial of the second kind.

1, -2, 3, 3, -10, 8, -4, 22, -38, 21, 5, -40, 111, -130, 55, -6, 65, -256, 474, -420, 144, 7, -98, 511, -1324, 1836, -1308, 377, -8, 140, -924, 3130, -6020, 6666, -3970, 987, 9, -192, 1554, -6588, 16435, -25088, 23109, -11822, 2584, -10, 255, -2472, 12720, -39430, 77645, -98160, 77378, -34690, 6765

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Sep 24 2006

Keywords

• sign
• tabl

Comments

The n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A053122(n,j)*x^j*(1 - x)^(n - j).

Examples

			Triangle begins:
     1;
    -2,    3;
     3,  -10,    8;
    -4,   22,   38,    21;
     5,  -40,  111,  -130,    55;
    -6,   65, -256,   474,  -420,    144;
     7,  -98,  511, -1324,  1836,  -1308,   377;
    -8,  140, -924,  3130, -6020,   6666, -3970,    987;
     9, -192, 1554, -6588, 16435, -25088, 23109, -11822, 2584;
     ... reformatted and extended. _Franck Maminirina Ramaharo_, Oct 10 2018
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the Second Kind
• Wikipedia, Chebyshev polynomials

Crossrefs

Cf. A008310, A049310, A053122, A111006.

Cf. A122753, A123018, A123019, A123021, A123199, A123202, A123217, A123221.

Programs

Mathematica

b0 = Table[CoefficientList[ChebyshevU[n, x/2 -1], x], {n, 0, 10}];
Table[CoefficientList[Sum[b0[[m+1]][[n+1]]*x^n*(1-x)^(m-n), {n,0,m}], x], {m,0,10}]//Flatten
(* Alternative Adamson Matrix method *)
t[n_, m_] = If[n==m, 2, If[n==m-1 || n==m+1, 1, 0]];
M[d_] := Table[t[n, m], {n, d}, {m, d}];
a = Join[{{1}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]];
Table[CoefficientList[Sum[a[[m+1]][[n+1]]*x^n*(1-x)^(m-n), {n, 0, m}], x], {m, 0, 10}]//Flatten

Maxima

A053122(n, k) := if n < k then 0 else ((-1)^(n - k))*binomial(n + k + 1,  2*k + 1)$
P(x, n) := expand(sum(A053122(n, j)*x^j*(1 - x)^(n - j), j, 0, n))$
T(n, k) := ratcoef(P(x, n), x, k)$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, hipow(P(x, n), x)))$ /* Franck Maminirina Ramaharo, Oct 10 2018 */

Sage

def A053122(n, k): return 0 if (nA053122(n, j)*x^j*(1-x)^(n-j) for j in (0..n) )
def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
flatten([T(n) for n in (0..12)]) # G. C. Greubel, Jul 15 2021

Formula

From Franck Maminirina Ramaharo, Oct 10 2018: (Start)

Row n = coefficients in the expansion of (1/sqrt((5*x - 4)*x))*(((3*x - 2 + sqrt((5*x - 4)*x))/2)^(n + 1) - ((3*x - 2 - sqrt((5*x - 4)*x))/2)^(n + 1)).

G.f.: 1/(1 + (2 - 3*x)*t + (1 - x)^2*t^2).

E.g.f.: exp(t*(3*x - 2)/2)*(sqrt((5*x - 4)*x)*cosh(t*sqrt((5*x - 4)*x)/2) + (3*x - 2)*sinh(t*sqrt((5*x - 4)*x)/2))/sqrt((5*x - 4)*x).

T(n,1) = (-1)^(n+1)*A006503(n).

T(n,n) = A001906(n+1). (End)

Extensions

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 10 2018