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

Author

Emeric Deutsch, Dec 04 2006

Keywords

• nonn
• tabf

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)

Examples

			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;
		

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.

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.

Programs

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 *)

Extensions

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

Original comment slightly edited by Antti Karttunen, Oct 27 2016