A213234 Triangle read by rows: coefficients of auxiliary Rudin-Shapiro polynomials A_{ns}(omega) written in descending powers of x.

2, 1, 1, -2, 1, -3, 1, -4, 2, 1, -5, 5, 1, -6, 9, -2, 1, -7, 14, -7, 1, -8, 20, -16, 2, 1, -9, 27, -30, 9, 1, -10, 35, -50, 25, -2, 1, -11, 44, -77, 55, -11, 1, -12, 54, -112, 105, -36, 2, 1, -13, 65, -156, 182, -91, 13, 1, -14, 77, -210, 294, -196, 49, -2, 1, -15, 90, -275, 450, -378, 140, -15

Offset: 0

Author

N. J. A. Sloane, Jun 06 2012

Keywords

• sign
• tabf

Comments

From Mohammed Yaseen, Nov 09 2024: (Start)

Let m + 1/m = x, then

m^2 + 1/m^2 = x^2 - 2,

m^3 + 1/m^3 = x^3 - 3*x,

m^4 + 1/m^4 = x^4 - 4*x^2 + 2,

m^5 + 1/m^5 = x^5 - 5*x^3 + 5*x,

m^6 + 1/m^6 = x^6 - 6*x^4 + 9*x^2 - 2,

m^7 + 1/m^7 = x^7 - 7*x^5 + 14*x^3 - 7*x, etc. (End)

Examples

			The first few polynomials are:
  2
  x
  x^2-2
  x^3-3*x
  x^4+2-4*x^2
  x^5-5*x^3+5*x
  x^6-2-6*x^4+9*x^2
  x^7-7*x^5+14*x^3-7*x
  x^8+2-8*x^6+20*x^4-16*x^2
  x^9-9*x^7+27*x^5-30*x^3+9*x
  x^10-2-10*x^8+35*x^6-50*x^4+25*x^2
  x^11-11*x^9+44*x^7-77*x^5+55*x^3-11*x
  x^12+2-12*x^10+54*x^8-112*x^6+105*x^4-36*x^2
  ...
Triangle begins:
  [2]
  [1]
  [1, -2]
  [1, -3]
  [1, -4, 2]
  [1, -5, 5]
  [1, -6, 9, -2]
  [1, -7, 14, -7]
  [1, -8, 20, -16, 2]
  [1, -9, 27, -30, 9]
  [1, -10, 35, -50, 25, -2]
  [1, -11, 44, -77, 55, -11]
  [1, -12, 54, -112, 105, -36, 2]
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10200 (rows 0 <= n <= 200, flattened)
• John Brillhart, John, J. S. Lomont, and Patrick Morton, Cyclotomic properties of the Rudin-Shapiro polynomials, J. Reine Angew. Math. 288 (1976), 37-65; see Table 2; MR0498479 (58 #16589).
• Matty van Son, Equations of the Cayley Surface, arXiv:2108.02441 [math.NT], 2021.

Crossrefs

Cf. A034807, A132460.

Programs

Maple

#The program is valid for n>=1:
f:=n->x^n+add((-1)^i*(n/i)*binomial(n-i-1,i-1)*x^(n-2*i), i=1..floor(n/2));
g:=n->series(x^n*subs(x=1/x,f(n)),x,n+1);
h:=n->seriestolist(series(subs(x=sqrt(x),g(n)),x,n+1));
for n from 0 to 15 do lprint(h(n)); od:

Mathematica

Block[{t}, t[0, 0] = 2; t[n_, k_] := Binomial[n - k, k] + Binomial[n - k - 1, k - 1]; Table[(-1)^k*t[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}] ] // Flatten (* Michael De Vlieger, Jun 26 2020, after Jean-François Alcover at A034807 *)

Formula

T(n,k) = (-1)^k*A034807(n,k). - Philippe Deléham , Nov 10 2013