A125184 - OEIS

%I #46 Sep 25 2019 03:31:52

%S 0,1,0,1,1,1,0,0,1,1,2,0,1,1,1,1,1,0,0,0,1,1,2,1,0,1,2,1,3,1,0,0,1,1,

%T 1,2,2,0,1,1,1,1,1,1,1,0,0,0,0,1,1,2,1,1,0,1,2,1,1,3,3,0,0,1,2,1,4,3,

%U 0,1,3,1,1,3,2,1,0,0,0,1,1,1,2,3,1,0,1,2,2,1,3,3,1,0,0,1,1,1,1,2,2,2,0,1,1,1

%N Triangle read by rows: T(n,k) is the coefficient of t^k in the Stern polynomial B(n,t) (n>=0, k>=0).

%C The Stern polynomials B(n,t) are defined by B(0,t)=0, B(1,t)=1, B(2n,t)=tB(n,t), B(2n+1,t)=B(n+1,t)+B(n,t) for n>=1 (see S. Klavzar et al.).

%C Also number of hyperbinary representations of n-1 containing exactly k digits 1. A hyperbinary representation of a nonnegative integer n is a representation of n as a sum of powers of 2, each power being used at most twice. Example: row 9 of the triangle is 1,2,1; indeed the hyperbinary representations of 8 are 200 (2*2^2+0*2^1+0*2^0), 120 (1*2^2+2*2^1+0*2^0), 1000 (1*2^3+0*2^2+0*2^1+0*2^0) and 112 (1*2^2+1*2^1+2*1^0), having 0,1,1 and 2 digits 1, respectively (see S. Klavzar et al. Corollary 3).

%C Number of terms in row n is A277329(n) (= 1+A057526(n) for n >= 1).

%C Row sums yield A002487 (Stern's diatomic series).

%C T(2n+1,1) = A005811(n) = number of 1's in the standard Gray code of n (S. Klavzar et al. Theorem 8). T(4n+1,1)=1, T(4n+3,1)=0 (S. Klavzar et al., Lemma 5).

%C From _Antti Karttunen_, Oct 27 2016: (Start)

%C Number of nonzero terms on row n is A277314(n).

%C Number of odd terms on row n is A277700(n).

%C Maximal term on row n is A277315(n).

%C Product of nonzero terms on row n is A277325(n).

%C Number of times where row n and n+1 both contain nonzero term in the same position is A277327(n).

%C (End)

%H T. D. Noe, <a href="/A125184/b125184.txt">Rows n = 0..1000, Flattened</a>

%H B. Adamczewski, <a href="http://dx.doi.org/10.4064/aa142-1-6">Non-converging continued fractions related to the Stern diatomic sequence</a>, Acta Arithm. 142 (1) (2010) 67-78.

%H N. Calkin and H. S. Wilf, <a href="http://www.math.upenn.edu/~wilf/website/recounting.pdf">Recounting the rationals</a>, Amer. Math. Monthly, 107 (No. 4, 2000), pp. 360-363.

%H K. Dilcher, L. Ericksen, <a href="http://dml.cz/handle/10338.dmlcz/143908">Reducibility and irreducibility of Stern (0, 1)-polynomials</a>, Communications in Mathematics, Volume 22/2014 , pp. 77-102.

%H K. Dilcher and K. B. Stolarsky, <a href="http://dx.doi.org/10.1016/j.aam.2006.01.003">A polynomial analogue to the Stern sequence</a>, Int. J. Number Theory 3 (1) (2007) 85-103.

%H S. Klavzar, U. Milutinovic and C. Petr, <a href="http://dx.doi.org/10.1016/j.aam.2006.01.003">Stern polynomials</a>, Adv. Appl. Math. 39 (2007) 86-95.

%H D. H. Lehmer, <a href="http://www.jstor.org/stable/2299356">On Stern's Diatomic Series</a>, Amer. Math. Monthly 36(1) 1929, pp. 59-67.

%H D. H. Lehmer, <a href="/A002487/a002487_1.pdf">On Stern's Diatomic Series</a>, Amer. Math. Monthly 36(1) 1929, pp. 59-67. [Annotated and corrected scanned copy]

%H Maciej Ulas, <a href="https://arxiv.org/abs/1909.10844">Strong arithmetic property of certain Stern polynomials</a>, arXiv:1909.10844 [math.NT], 2019.

%H Maciej Ulas and Oliwia Ulas, <a href="http://arxiv.org/abs/1102.5109">On certain arithmetic properties of Stern polynomials</a>, arXiv:1102.5109 [math.CO], 2011.

%e Triangle starts:

%e 0;

%e 1;

%e 0, 1;

%e 1, 1;

%e 0, 0, 1;

%e 1, 2;

%e 0, 1, 1;

%e 1, 1, 1;

%e 0, 0, 0, 1;

%e 1, 2, 1;

%e 0, 1, 2;

%e 1, 3, 1;

%p B:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then t*B(n/2) else B((n+1)/2)+B((n-1)/2) fi end: for n from 0 to 36 do B(n):=sort(expand(B(n))) od: dg:=n->degree(B(n)): 0; for n from 0 to 40 do seq(coeff(B(n),t,k),k=0..dg(n)) od; # yields sequence in triangular form

%t B[0, _] = 0; B[1, _] = 1; B[n_, t_] := B[n, t] = If[EvenQ[n], t*B[n/2, t], B[1 + (n-1)/2, t] + B[(n-1)/2, t]]; row[n_] := CoefficientList[B[n, t], t]; row[0] = {0}; Array[row, 40, 0] // Flatten (* _Jean-François Alcover_, Jul 30 2015 *)

%Y Cf. A057526, A002487, A005811.

%Y Cf. A186890 (n such that the Stern polynomial B(n,x) is self-reciprocal).

%Y Cf. A186891 (n such that the Stern polynomial B(n,x) is irreducible).

%Y Cf. A260443 (Stern polynomials encoded in the prime factorization of n).

%Y Cf. also A277314, A277315, A277325, A277327, A277329, A277700.

%K nonn,tabf

%O 0,11

%A _Emeric Deutsch_, Dec 04 2006

%E 0 prepended by _T. D. Noe_, Feb 28 2011

%E Original comment slightly edited by _Antti Karttunen_, Oct 27 2016