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A267486
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Triangle of coefficients of Gaussian polynomials [2n+7,6]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=6n+3.
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6
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-1, -2, 1, 1, 0, 2, -2, -15, 7, 17, -5, -7, 1, 1, -2, -6, 25, 71, -80, -218, 126, 284, -106, -190, 48, 69, -11, -13, 1, 1, 0, 6, -12, -137, 196, 945, -811, -2745, 1602, 4163, -1780, -3711, 1193, 2059, -493, -722, 123, 156, -17, -19, 1, 1, -3, -12, 94, 358, -952, -3430, 4699, 15615, -13467, -39946, 24494, 63168, -29535, -65638, 24206, 46512, -13652, -22891, 5294, 7834, -1386, -1831, 234, 279, -23, -25, 1, 1
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OFFSET
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0,2
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COMMENTS
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The entry a(n,k), n >= 0, k = 0,1,...,g, where g=6n+3, of this irregular triangle is the coefficient of (1+q^2)^k*q^(g-k) in the representation of the Gaussian polynomial [2n+7,6]_q = Sum_{k=0..g) a(n,k)*(1+q^2)^k*q^(g-k).
Row n is of length 6n+4.
The sequence arises in the formal derivation of the stability polynomial B(x) = Sum_{i=0..N} d_i T(iM,x) of rank N, and degree L, where T(iM,x) denotes the Chebyshev polynomial of the first kind of degree iM. The coefficients d_i are determined by order conditions on the stability polynomial.
Conjecture: More generally, the Gaussian polynomial [2*n+m+1-(m mod 2),m]_q = Sum_{k=0..g(m;n)} a(m;n,k)*(1+q^2)^k*q^(g(m;n)-k), for m >= 0, n >= 0, where g(m;n) = m*n if m is odd and (2*n+1)*m/2 if m is even, and the tabf array entries a(m;n,k) are the coefficients of the g.f. for the row n polynomials G(m;n,x) = (d^m/dt^m)G(m;n,t,x)/m!|_{t=0}, with G(m;n,t,x) = (1+t)*Product_{k=1..n+(m - m (mod 2))/2}(1 + t^2 + 2*t*T(k,x/2) (Chebyshev's T-polynomials). Hence a(m;n,k) = [x^k]G(m;n,x), for k=0..g(m;n). The present entry is the instance m = 2. (Thanks to Wolfdieter Lang for clarifying the text on the general prescription of a(m;n,k).)
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LINKS
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FORMULA
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G.f. for row polynomial: G(n,x) = (d^6/dt^6)((1+t)*Product_{i=1..n+1}(1+t^2+2t*T(i,x/2))/6!)|_{t=0}.
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EXAMPLE
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-1,-2,1,1;
0,2,-2,-15,7,17,-5,-7,1,1;
-2,-6,25,71,-80,-218,126,284,-106,-190,48,69,-11,-13,1,1;
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MAPLE
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A267486 := proc (n, k) local y: y := expand(subs(t = 0, diff((1+t)*product(1+t^2+2*t*ChebyshevT(i, x/2), i = 1 .. n+3), t$6)/6!)): if k = 0 then subs(x = 0, y) else subs(x = 0, diff(y, x$k)/k!) end if: end proc: seq(seq(A267486(n, k), k = 0 .. 6*n+3), n = 0 .. 20);
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MATHEMATICA
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row[n_] := 1/6! D[(1+t)*Product[1+t^2+2*t*ChebyshevT[i, x/2], {i, 1, n+1}], {t, 6}] /. t -> 0 // CoefficientList[#, x]&; Table[row[n], {n, 0, 20}] // Flatten (* From A267120 entry by Jean-François Alcover *)
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CROSSREFS
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KEYWORD
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sign,tabf
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AUTHOR
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EXTENSIONS
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Added row length
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STATUS
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approved
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