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).
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
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