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 A128505 Irregular triangular array a(n,m) for third (k=3) convolution of Chebyshev's S(n,x) = U(n,x/2) polynomials, read by rows (n >=0, 0 <= m <= floor(n/2)). 2
 1, 4, 10, -4, 20, -20, 35, -60, 10, 56, -140, 60, 84, -280, 210, -20, 120, -504, 560, -140, 165, -840, 1260, -560, 35, 220, -1320, 2520, -1680, 280, 286, -1980, 4620, -4200, 1260, -56, 364, -2860, 7920, -9240, 4200, -504, 455, -4004, 12870, -18480, 11550, -2520, 84, 560, -5460, 20020, -34320 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS S3(n,x) := Sum_{k=0..n} S(n-k,x)*S2(k,x) = Sum_{m=0..floor(n/2)} a(n,m)*x^(n-2*m)  with the second convolution S2(n,x) given by array A128503. Row polynomials P3(n,x) :=  Sum_{m=0..floor(n/2)} a(n,m)*x^m (increasing powers of x). LINKS Wolfdieter Lang, First 15 rows and more. FORMULA a(n,m) = binomial(n-m+3,3)*binomial(n-m,m)*(-1)^m, m = 0..floor(n/2), n >= 0. a(n,m) = binomial(m+3,3)*binomial(n-m+3,m+3)*(-1)^m, m = 0..floor(n/2), n >= 0. G.f. for S3(n,x): 1/(1-x*z+z^2)^4. G.f. for P3(n,x): 1/(1-z+x*z^2)^4. EXAMPLE 1;   4;   10,   -4;   20,  -20;   35,  -60,  10;   56, -140,  60;   84, -280, 210,  -20;   120,-504, 560, -140;   ... n=4: [35,-60,10] stands also for the row polynomial P3(4,x) = 35-60*x+10*x^2. CROSSREFS Row sums (signed array) give A128506. Unsigned row sums are A001872. Cf. A128503 (k=2 convolution). Sequence in context: A151707 A059132 A059136 * A336988 A200454 A303052 Adjacent sequences:  A128502 A128503 A128504 * A128506 A128507 A128508 KEYWORD sign,tabf,easy AUTHOR Wolfdieter Lang, Apr 04 2007 EXTENSIONS Name edited by Petros Hadjicostas, Sep 04 2019 STATUS approved

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Last modified June 16 04:56 EDT 2021. Contains 345056 sequences. (Running on oeis4.)