A125184 Triangle read by rows: T(n,k) is the coefficient of t^k in the Stern polynomial B(n,t) (n>=0, k>=0).

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, 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, 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

Offset

0,11

Comments

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.).

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).

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

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

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).

From Antti Karttunen, Oct 27 2016: (Start)

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

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

Maximal term on row n is A277315(n).

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

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

(End)

Links

T. D. Noe, Rows n = 0..1000, Flattened

B. Adamczewski, Non-converging continued fractions related to the Stern diatomic sequence, Acta Arithm. 142 (1) (2010) 67-78.

N. Calkin and H. S. Wilf, Recounting the rationals, Amer. Math. Monthly, 107 (No. 4, 2000), pp. 360-363.

K. Dilcher, L. Ericksen, Reducibility and irreducibility of Stern (0, 1)-polynomials, Communications in Mathematics, Volume 22/2014 , pp. 77-102.

K. Dilcher and K. B. Stolarsky, A polynomial analogue to the Stern sequence, Int. J. Number Theory 3 (1) (2007) 85-103.

S. Klavzar, U. Milutinovic and C. Petr, Stern polynomials, Adv. Appl. Math. 39 (2007) 86-95.

D. H. Lehmer, On Stern's Diatomic Series, Amer. Math. Monthly 36(1) 1929, pp. 59-67.

D. H. Lehmer, On Stern's Diatomic Series, Amer. Math. Monthly 36(1) 1929, pp. 59-67. [Annotated and corrected scanned copy]

Maciej Ulas, Strong arithmetic property of certain Stern polynomials, arXiv:1909.10844 [math.NT], 2019.

Maciej Ulas and Oliwia Ulas, On certain arithmetic properties of Stern polynomials, arXiv:1102.5109 [math.CO], 2011.

Example

Triangle starts:
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;

Maple

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

Mathematica

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 *)

Crossrefs

Cf. A057526, A002487, A005811.

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

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

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

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

Sequence in context: A025913 A123230 A078821 * A236575 A059282 A114591

Adjacent sequences: A125181 A125182 A125183 * A125185 A125186 A125187

Keyword

nonn,tabf

Author

Emeric Deutsch, Dec 04 2006

Extensions

0 prepended by T. D. Noe, Feb 28 2011

Original comment slightly edited by Antti Karttunen, Oct 27 2016

Status

approved