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.

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, -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, -14, 12, -10, 8, -3, -6, -1, 1, 11, 35, 21, -22, 14, -10, 10, -11, 7, 7, 1

Offset

0,5

Comments

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).

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.

Links

Kevin Ryde, Table of n, a(n) for rows 0 to 100, flattened

John Brillhart and Patrick Morton, Über Summen von Rudin-Shapiroschen Koeffizienten, Illinois Journal of Mathematics, volume 22, issue 1, 1978, pages 126-148.

Christophe Doche and Michel Mendès France, An Exercise on the Average Number of Real Zeros of Random Real Polynomials, Finite and Infinite Combinatorics conference, Budapest, 2001, pages 1-14, see Rudin-Shapiro example page 9.

Formula

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.

T(n,k) = (-1)^k * A104967(n,n-k).

Row polynomial P_n(y) = (1-y)*P_{n-1}(y) + 2*y*P_{n-2}(y) for n>=2. [Doche and Mendès France]

G.f.: (1 + 2*x*y)/(1 + x*(y-1) - 2*x^2*y).

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.

Example

Triangle begins
        k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7
  n=0:   1
  n=1:   1,  1
  n=2:   1,  2, -1
  n=3:   1,  3, -1,  1
  n=4:   1,  4,  0,  0, -1
  n=5:   1,  5,  2, -2,  1,  1
  n=6:   1,  6,  5, -4,  3, -2, -1
  n=7:   1,  7,  9, -5,  3, -3,  3,  1
For T(5,3), those j in the range 0 <= j < 2^5 with wt(j) = 3 are
  j      =  7 11 13 14 19 21 22 25 26 28
  GRS(j) = +1 -1 -1 +1 -1 +1 -1 -1 -1 +1 total -2 = T(5,3)

Prog

(PARI) my(M=Mod('x, 'x^2-(1-'y)*'x-2*'y)); row(n) = Vecrev(subst(lift(M^n), 'x, 'y+1));

Crossrefs

Cf. A020985 (GRS), A020986 (GRS partial sums), A000120 (binary weight), A285869.

Columns k=0..3: A000012, A001477, A000096, A275874.

Cf. A165326 (main diagonal), A248157 (second diagonal negated).

Cf. A060546 (row sums), A104969 (row sums squared terms).

Cf. A329301 (antidiagonal sums).

Cf. A104967 (rows reversed, up to signs).

Sequence in context: A196931 A175465 A080209 * A127949 A051340 A167407

Adjacent sequences: A347168 A347169 A347170 * A347172 A347173 A347174

Keyword

sign,look,tabl

Author

Kevin Ryde, Aug 21 2021

Status

approved