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A347171 Triangle read by rows where T(n,k) is the sum of Golay-Rudin-Shapiro terms GRS(j) (A020985) for j in the range 0 <= j < 2^n and having binary weight wt(j) = A000120(j) = k.

%I A347171 #15 Sep 06 2021 04:50:43
%S A347171 1,1,1,1,2,-1,1,3,-1,1,1,4,0,0,-1,1,5,2,-2,1,1,1,6,5,-4,3,-2,-1,1,7,9,
%T A347171 -5,3,-3,3,1,1,8,14,-4,0,0,2,-4,-1,1,9,20,0,-6,6,-4,0,5,1,1,10,27,8,
%U A347171 -14,12,-10,8,-3,-6,-1,1,11,35,21,-22,14,-10,10,-11,7,7,1
%N A347171 Triangle read by rows where T(n,k) is the sum of Golay-Rudin-Shapiro terms GRS(j) (A020985) for j in the range 0 <= j < 2^n and having binary weight wt(j) = A000120(j) = k.
%C A347171 Doche and Mendès France form polynomials P_n(y) = Sum_{j=0..2^n-1} GRS(j) * y^wt(j) and here row n is the coefficients of P_n starting from the constant term, so P_n(y) = Sum_{k=0..n} T(n,k)*y^k.  They conjecture that the number of real roots of P_n is A285869(n).
%C A347171 Row sum n is the sum of GRS terms from j = 0 to 2^n-1 inclusive, which Brillhart and Morton (Beispiel 6 page 129) show is A020986(2^n-1) = 2^ceiling(n/2) = A060546(n).  The same follows by substituting y=1 in the P_n recurrence or the generating function.
%H A347171 Kevin Ryde, <a href="/A347171/b347171.txt">Table of n, a(n) for rows 0 to 100, flattened</a>
%H A347171 John Brillhart and Patrick Morton, <a href="http://projecteuclid.org/euclid.ijm/1256048841">Über Summen von Rudin-Shapiroschen Koeffizienten</a>, Illinois Journal of Mathematics, volume 22, issue 1, 1978, pages 126-148.
%H A347171 Christophe Doche and Michel Mendès France, <a href="http://web.science.mq.edu.au/~doche/finf.ps">An Exercise on the Average Number of Real Zeros of Random Real Polynomials</a>, Finite and Infinite Combinatorics conference, Budapest, 2001, pages 1-14, see Rudin-Shapiro example page 9.
%F A347171 T(n,k) = T(n-1,k) - T(n-1,k-1) + 2*T(n-2,k-1) for n>=2, and taking T(n,k)=0 if k<0 or k>n.
%F A347171 T(n,k) = (-1)^k * A104967(n,n-k).
%F A347171 Row polynomial P_n(y) = (1-y)*P_{n-1}(y) + 2*y*P_{n-2}(y) for n>=2. [Doche and Mendès France]
%F A347171 G.f.: (1 + 2*x*y)/(1 + x*(y-1) - 2*x^2*y).
%F A347171 Column g.f.: C_k(x) = 1/(1-x) for k=0 and C_k(x) = x^k * (2*x-1)^(k-1) / (1-x)^(k+1) for k>=1.
%e A347171 Triangle begins
%e A347171         k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7
%e A347171   n=0:   1
%e A347171   n=1:   1,  1
%e A347171   n=2:   1,  2, -1
%e A347171   n=3:   1,  3, -1,  1
%e A347171   n=4:   1,  4,  0,  0, -1
%e A347171   n=5:   1,  5,  2, -2,  1,  1
%e A347171   n=6:   1,  6,  5, -4,  3, -2, -1
%e A347171   n=7:   1,  7,  9, -5,  3, -3,  3,  1
%e A347171 For T(5,3), those j in the range 0 <= j < 2^5 with wt(j) = 3 are
%e A347171   j      =  7 11 13 14 19 21 22 25 26 28
%e A347171   GRS(j) = +1 -1 -1 +1 -1 +1 -1 -1 -1 +1 total -2 = T(5,3)
%o A347171 (PARI) my(M=Mod('x, 'x^2-(1-'y)*'x-2*'y)); row(n) = Vecrev(subst(lift(M^n),'x,'y+1));
%Y A347171 Cf. A020985 (GRS), A020986 (GRS partial sums), A000120 (binary weight), A285869.
%Y A347171 Columns k=0..3: A000012, A001477, A000096, A275874.
%Y A347171 Cf. A165326 (main diagonal), A248157 (second diagonal negated).
%Y A347171 Cf. A060546 (row sums), A104969 (row sums squared terms).
%Y A347171 Cf. A329301 (antidiagonal sums).
%Y A347171 Cf. A104967 (rows reversed, up to signs).
%K A347171 sign,look,tabl
%O A347171 0,5
%A A347171 _Kevin Ryde_, Aug 21 2021