A128932 -id:A128932 - cp's OEIS frontend

A335444 Define the Fibonacci polynomials by F[1] = 1, F[2] = x; for n > 2, F[n] = xF[n-1] + F[n-2] (cf. A049310, A053119). Swamy's inequality states that F[n]^2 <= G[n] = (x^2 + 1)^2(x^2 + 2)^(n-3) for all n >= 3 and all real x. The sequence gives a triangle of the coefficients of the even exponents of G[n] - F[n]^2 read by rows.

0, 2, 1, 3, 6, 2, 8, 19, 14, 3, 15, 52, 58, 26, 4, 32, 128, 192, 132, 42, 5, 63, 300, 558, 518, 253, 62, 6, 128, 679, 1496, 1742, 1152, 433, 86, 7, 255, 1506, 3801, 5294, 4413, 2248, 684, 114, 8, 512, 3292, 9308, 14999, 15040, 9680, 3992, 1018, 146, 9

Offset: 3

Petros Hadjicostas, Jun 10 2020

Swamy's (1966) inequality states that F[n]^2 <= G[n] for all real x and all integers n >= 3.
Row n >= 3 of this irregular table gives the coefficients of the even powers of the polynomial G[n] - F[n]^2 (with exponents in increasing order). The coefficients of the odd powers are zero, and they are thus omitted. The degree of G[n] - F[n]^2 is 2*n - 6, so row n >= 3 contains n - 2 terms.
To prove that the degree of G[n] - F[n]^2 is 2*n - 6, note that the first few terms of G[n] are x^(2*n-2) + 2*(n-2)*x^(2*n-4) + (2*n^2 - 10*n + 13)*x^(2*n-6) + ... while the first few terms of F[n]^2 are x^(2*n-2) + 2*(n-2)*x^(2*n-4) + (2*n^2 - 11*n + 16)*x^(2*n-6) + ..., so the leading term of the polynomial G[n] - F[n]^2 is (n-3)*x^(2*n-6).
Guilfoyle (1967) notes that F[n] = det(A_n), where A_n is the (n-1) X (n-1) matrix [[x, -1, 0, 0, ..., 0, 0, 0], [1, x, -1, 0, ..., 0, 0, 0], [0, 1, x, -1, ..., 0, 0, 0], ..., [0, 0, 0, 0, ..., 1, x, -1], [0, 0, 0, 0, ..., 0, 1, x]], and Swamy's original inequality follows from Hadamard's inequality.
Koshy (2019) writes Swamy's original inequality in the form x^(n-3)*F[n]^2 <= F[3]^2*F[4]^(n-3) for x >= 1, and gives a counterpart inequality for Lucas polynomials. Notice, however, that the original form of Swamy's inequality is true for all real x.

			Triangle T(n,k) (with rows n >= 3 and columns k = 0..n-3) begins:
   0;
   2,   1;
   3,   6,   2;
   8,  19,  14,   3;
  15,  52,  58,  26,  4;
  32, 128, 192, 132, 42, 5;
  ...

Thomas Koshy, Fibonacci and Lucas numbers with Applications, Vol. 2, Wiley, 2019; see p. 33. [Vol. 1 was published in 2001.]
D. S. Mitrinovic, Analytic Inequalities, Springer-Verlag, 1970; see p. 232, Sect. 3.3.38.

Richard Guilfoyle, Comment to the solution of Problem E1846, Amer. Math. Monthly, 74(5), 1967, 593. [It is pointed out that the inequality is a special case of Hadamard's inequality.]
M. N. S. Swamy, Problem E1846 proposed for solution, Amer. Math. Monthly, 73(1) (1966), 81.
M. N. S. Swamy and R. E. Giudici, Solution to Problem E1846, Amer. Math. Monthly, 74(5), 1967, 592-593.
M. N. S. Swamy and Joel Pitcain, Comment to Problem E1846, Amer. Math. Monthly, 75(3) (1968), 295. [It is pointed out that I^{n-1}*F[n](x) = U_{n-1}(I*x/2), where U_{n-1}(cos(t)) = sin(n*t)/sin(t) and I = sqrt(-1); Cf. A049310 and A053119, but with different notation.]
Wikipedia, Fibonacci polynomials.
Wikipedia, Hadamard's inequality.

Cf. A002620, A045623, A049310, A051890, A053119, A128932 (similar), A166920.

lista(nn) = {my(f=vector(nn)); my(g=vector(nn)); my(h=vector(nn)); f[1]=1; f[2]=x; g[1]=0; g[2]=0; for(n=3, nn, g[n] = (x^2+1)^2*(x^2+2)^(n-3)); for(n=3, nn, f[n] = x*f[n-1]+f[n-2]); for(n=1, nn, h[n] = g[n]-f[n]^2); for(n=3, nn, for(k=0, n-3, print1(polcoef(h[n], 2*k, x), ", ")); print(); ); }

T(n,0) = 2^(n-3) - (1 - (-1)^n)/2 = A166920(n-3) for n >= 3.
T(n,1) = 2^(n-4)*(n + 1) - floor(n/2)*ceiling(n/2) = A045623(n-2) - A002620(n) for n >= 4.
T(n, n-4) = 2*(n^2 - 7*n + 13) = A051890(n-3) for n >= 4.
T(n, n-3) = n - 3 for n >= 3.

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