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
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