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

%I A213234 #36 Nov 24 2024 10:10:31
%S A213234 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,
%T A213234 27,-30,9,1,-10,35,-50,25,-2,1,-11,44,-77,55,-11,1,-12,54,-112,105,
%U A213234 -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
%N A213234 Triangle read by rows: coefficients of auxiliary Rudin-Shapiro polynomials A_{ns}(omega) written in descending powers of x.
%C A213234 From _Mohammed Yaseen_, Nov 09 2024: (Start)
%C A213234 Let m   + 1/m   = x, then
%C A213234     m^2 + 1/m^2 = x^2 - 2,
%C A213234     m^3 + 1/m^3 = x^3 - 3*x,
%C A213234     m^4 + 1/m^4 = x^4 - 4*x^2 + 2,
%C A213234     m^5 + 1/m^5 = x^5 - 5*x^3 + 5*x,
%C A213234     m^6 + 1/m^6 = x^6 - 6*x^4 + 9*x^2 - 2,
%C A213234     m^7 + 1/m^7 = x^7 - 7*x^5 + 14*x^3 - 7*x, etc. (End)
%H A213234 Michael De Vlieger, <a href="/A213234/b213234.txt">Table of n, a(n) for n = 0..10200</a> (rows 0 <= n <= 200, flattened)
%H A213234 John Brillhart, John, J. S. Lomont, and Patrick Morton, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002192802">Cyclotomic properties of the Rudin-Shapiro polynomials</a>, J. Reine Angew. Math. 288 (1976), 37-65; see Table 2; MR0498479 (58 #16589).
%H A213234 Matty van Son, <a href="https://arxiv.org/abs/2108.02441">Equations of the Cayley Surface</a>, arXiv:2108.02441 [math.NT], 2021.
%F A213234 T(n,k) = (-1)^k*A034807(n,k). - _Philippe Deléham_ , Nov 10 2013
%e A213234 The first few polynomials are:
%e A213234   2
%e A213234   x
%e A213234   x^2-2
%e A213234   x^3-3*x
%e A213234   x^4+2-4*x^2
%e A213234   x^5-5*x^3+5*x
%e A213234   x^6-2-6*x^4+9*x^2
%e A213234   x^7-7*x^5+14*x^3-7*x
%e A213234   x^8+2-8*x^6+20*x^4-16*x^2
%e A213234   x^9-9*x^7+27*x^5-30*x^3+9*x
%e A213234   x^10-2-10*x^8+35*x^6-50*x^4+25*x^2
%e A213234   x^11-11*x^9+44*x^7-77*x^5+55*x^3-11*x
%e A213234   x^12+2-12*x^10+54*x^8-112*x^6+105*x^4-36*x^2
%e A213234   ...
%e A213234 Triangle begins:
%e A213234   [2]
%e A213234   [1]
%e A213234   [1, -2]
%e A213234   [1, -3]
%e A213234   [1, -4, 2]
%e A213234   [1, -5, 5]
%e A213234   [1, -6, 9, -2]
%e A213234   [1, -7, 14, -7]
%e A213234   [1, -8, 20, -16, 2]
%e A213234   [1, -9, 27, -30, 9]
%e A213234   [1, -10, 35, -50, 25, -2]
%e A213234   [1, -11, 44, -77, 55, -11]
%e A213234   [1, -12, 54, -112, 105, -36, 2]
%e A213234   ...
%p A213234 #The program is valid for n>=1:
%p A213234 f:=n->x^n+add((-1)^i*(n/i)*binomial(n-i-1,i-1)*x^(n-2*i), i=1..floor(n/2));
%p A213234 g:=n->series(x^n*subs(x=1/x,f(n)),x,n+1);
%p A213234 h:=n->seriestolist(series(subs(x=sqrt(x),g(n)),x,n+1));
%p A213234 for n from 0 to 15 do lprint(h(n)); od:
%t A213234 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 *)
%Y A213234 Cf. A034807, A132460.
%K A213234 sign,tabf
%O A213234 0,1
%A A213234 _N. J. A. Sloane_, Jun 06 2012