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

0,2

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).

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

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)

Example

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

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

Prog

(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 (n<k) else (-1)^(n-k)*binomial(n+k+1, 2*k+1)
def p(n, x): return sum( A053122(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

Crossrefs

Cf. A008310, A049310, A053122, A111006.

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

Sequence in context: A306101 A364967 A337432 * A100652 A094416 A218868

Adjacent sequences: A123024 A123025 A123026 * A123028 A123029 A123030

Keyword

sign,tabl

Author

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

Extensions

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

Status

approved