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Revision History for A332420

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A332420

Number of Maclaurin polynomials p(2m-1,x) of sin(x) having exactly n positive zeros.
(history; published version)

#11 by Michel Marcus at Tue Jan 21 23:55:41 EST 2025

STATUS

reviewed

approved

#10 by Joerg Arndt at Tue Jan 21 23:53:52 EST 2025

STATUS

proposed

reviewed

#9 by Jinyuan Wang at Tue Jan 21 21:24:13 EST 2025

STATUS

editing

proposed

#8 by Jinyuan Wang at Tue Jan 21 21:23:00 EST 2025

COMMENTS

Maclaurin polynomial p(2m-1,x) of sin(x) is x - x^3/3! + x^5/5! - ... - (-1)^m*x^(2m-1)/(2m-1)!.

CROSSREFS

Cf. A012265, A332325, A332417.

Discussion

Tue Jan 21

21:24

Jinyuan Wang: comment is also copied from A332325

#7 by Jinyuan Wang at Tue Jan 21 21:15:53 EST 2025

NAME

Number of Maclaurin polynomials p(2m-1,x) of sin (x ) having exactly n positive zeros.

DATA

3, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 5, 4

COMMENTS

a(1) counts these values of 2n-1: 3, 5, and 11. The single zeros of p(3,x), p(5,x), and p(11,x) are sqrt(6), 3.078642..., and 3.141148..., respectively.

EXAMPLE

a(1) counts these values of 2m-1: 3, 5, and 11. The single zeros of p(3,x), p(5,x), and p(11,x) are sqrt(6), 3.078642..., and 3.141148..., respectively.

MATHEMATICA

v = Table[Length[u[n]], {n, 2, 30100, 2}]

(1/2) Table[Count[v, n], {n, 1, 4010}]

CROSSREFS

Cf. A332417, A012265, A332325, A332417.

KEYWORD

nonn,hard,more

EXTENSIONS

More terms from Jinyuan Wang, Jan 21 2025

STATUS

approved

editing

Discussion

Tue Jan 21

21:20

Jinyuan Wang: name and Mma is consistent with A332325

#6 by N. J. A. Sloane at Mon Mar 09 15:11:56 EDT 2020

STATUS

proposed

approved

#5 by Clark Kimberling at Sat Mar 07 17:30:56 EST 2020

STATUS

editing

proposed

#4 by Clark Kimberling at Sat Mar 07 17:28:44 EST 2020

KEYWORD

nonn,easy,hard,more

STATUS

proposed

editing

#3 by Clark Kimberling at Thu Feb 13 09:55:19 EST 2020

STATUS

editing

proposed

Discussion

Fri Mar 06

23:11

N. J. A. Sloane: Both "easy" and "more" again?

#2 by Clark Kimberling at Thu Feb 13 09:52:01 EST 2020

NAME

allocated for Clark KimberlingNumber of Maclaurin polynomials of sin x having exactly n positive zeros.

DATA

3, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5

OFFSET

1,1

COMMENTS

a(1) counts these values of 2n-1: 3, 5, and 11. The single zeros of p(3,x), p(5,x), and p(11,x) are sqrt(6), 3.078642..., and 3.141148..., respectively.

MATHEMATICA

z = 60; p[n_, x_] := Normal[Series[Sin[x], {x, 0, n}]];

t[n_] := x /. NSolve[p[n, x] == 0, x, z];

u[n_] := Select[t[n], Im[#] == 0 && # > 0 &];

v = Table[Length[u[n]], {n, 2, 30}]

(1/2) Table[Count[v, n], {n, 1, 40}]

CROSSREFS

Cf. A332417, A332325.

KEYWORD

allocated

nonn,easy,more

AUTHOR

Clark Kimberling, Feb 13 2020

STATUS

approved

editing