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 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. 13
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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[ExpandAll[ChebyshevU[n, x/2 - 1]], x], {n, 0, 10}]; c0 = Table[CoefficientList[Sum[b0[[m + 1]][[n + 1]]*x^n*(1 - x)^(m - n), {n, 0, m}], x], {m, 0, 10}]; Flatten[c0] (* 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, 1, d}, {m, 1, d}]; Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}]; a = Join[{{1}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; b = Table[CoefficientList[Sum[a[[m + 1]][[n + 1]]*x^n*(1 - x)^(m - n), {n, 0, m}], x], {m, 0, 10}]; Flatten[b] 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 */ CROSSREFS Cf. A008310, A049310, A053122, A111006. Cf. A122753, A123018, A123019, A123021, A123199, A123202, A123202, A123217, A123221. Sequence in context: A194232 A110042 A306101 * 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

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Last modified January 26 05:10 EST 2020. Contains 331273 sequences. (Running on oeis4.)