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A125184
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Triangle read by rows: T(n,k) is the coefficient of t^k in the Stern polynomial B(n,t) (n>=0, k>=0). 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.).
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11
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,11
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COMMENTS
| 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 1+A057526(n) (n>=2). 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).
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REFERENCES
| N. Calkin and H. S. Wilf, Recounting the rationals, Amer. Math. Monthly, 107 (No. 4, 2000), pp. 360-363.
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.
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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. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 07 2010]
Maciej Ulas and Oliwia Ulas, On certain arithmetic properties of Stern polynomials (arXiv:1102.5109)
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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;
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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
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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).
Sequence in context: A025913 A123230 A078821 * A091430 A059282 A161849
Adjacent sequences: A125181 A125182 A125183 * A125185 A125186 A125187
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 04 2006
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EXTENSIONS
| 0 prepended by T. D. Noe (noe(AT)sspectra.com), Feb 28 2011
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