A199220 -id:A199220 - cp's OEIS frontend

A199221 Triangle read by rows: T(n,k) = (n+1-k)|s(n,n+1-k)| - 2|s(n-1,n-k)|, where s(n,k) are the signed Stirling numbers of the first kind and 1 <= k <= n.

-1, 0, 1, 1, 4, 2, 2, 12, 18, 6, 3, 28, 83, 88, 24, 4, 55, 270, 575, 500, 120, 5, 96, 705, 2490, 4324, 3288, 720, 6, 154, 1582, 8330, 23828, 35868, 24696, 5040, 7, 232, 3178, 23296, 98707, 242872, 328236, 209088, 40320, 8, 333, 5868, 57078, 334740, 1212057, 2658472, 3298932, 1972512, 362880, 9, 460, 10140, 126300, 977865, 4873680, 15637290, 31292600, 36207576, 20531520, 3628800

Offset: 1

Frank M Jackson, Nov 04 2011

Use the T(n,k) as coefficients to generate a polynomial of degree n-1 in d as Sum_{k=1..n} T(n,k)d^(k-1) and let f(n) be the greatest root of this polynomial. Then a polygon of n sides that form a harmonic progression in the ratio 1 : 1/(1+d) : 1/(1+2d) : ... : 1/(1+(n-1)d) can only exist if the common difference d of the denominators is limited to the range f(n) < d < g(n). The higher limit g(n) is the greatest root of another group of polynomials defined by coefficients in the triangle A199220.

			Triangle starts:
  -1;
   0,  1;
   1,  4,   2;
   2, 12,  18,   6;
   3, 28,  83,  88,  24;
   4, 55, 270, 575, 500, 120;

Cf. A094638, A192918, A199220.

Flatten[Table[(n+1-k)Abs[StirlingS1[n,n+1-k]]-2Abs[StirlingS1[n-1,n-k]],{n,1,20},{k,1,n}]]

T(n,k) = (n+1-k)*abs(stirling(n,n+1-k,1)) - 2*abs(stirling(n-1,n-k,1));
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Sep 30 2018

The triangle of coefficients can be generated by expanding the equation (Sum_{k=1..n} 1/(1+(k-1)d)) - 2/(1+(n-1)d) = 0 into a polynomial of degree n-1 in d.

A199589 Decimal expansion of the greatest root of 6x^3 - 6x - 2 = 0.

Original entry on oeis.org

1, 1, 3, 7, 1, 5, 8, 0, 4, 2, 6, 0, 3, 2, 5, 7, 6, 1, 2, 8, 3, 7, 6, 6, 7, 9, 5, 1, 9, 2, 0, 0, 9, 8, 7, 6, 2, 5, 8, 1, 3, 6, 0, 3, 9, 4, 2, 2, 9, 9, 0, 6, 5, 8, 5, 9, 6, 2, 8, 8, 7, 9, 6, 4, 9, 4, 4, 2, 5, 1, 0, 6, 6, 5, 6, 8, 5, 0, 9, 4, 5, 4, 9, 8, 5, 3, 1, 6, 7, 7, 7, 6, 7, 8, 9, 9, 7, 7, 0

Offset: 1

Frank M Jackson, Nov 08 2011

If the side lengths of a quadrilateral form a harmonic progression in the ratio 1 : 1/(1+d) : 1/(1+2d) : 1/(1+3d) where d is the common difference between the denominators of the harmonic progression, then the triangle inequality condition requires that d be in the range f < d < g, where g = 1.1371580... and is the greatest root of the equation: 2 + 6d - 6d^3 = 0. The value of f is given in A199590.

			1.13715804260325761283766795192009876258136039422990658596288796494425...

Index entries for algebraic numbers, degree 3.

Cf. A010503, A016051, A199220, A199221, A199590.

N[Reduce[2+6d-6d^3==0, d], 100]
RealDigits[(2/Sqrt[3]) * Cos[Pi/18], 10, 120][[1]] (* Amiram Eldar, Nov 22 2024 *)

real(polroots(6*x^3-6*x-2)[3]) \\ Charles R Greathouse IV, Nov 10 2011

polrootsreal(6*x^3-6*x-2)[3] \\ Charles R Greathouse IV, Apr 14 2014

Equals sqrt(4/3)*cos(Pi/18). - Charles R Greathouse IV, Nov 10 2011

Equals Product_{k>=1} (1 - (-1)^k/A016051(k)). - Amiram Eldar, Nov 22 2024

A199590 Decimal expansion (unsigned) of the greatest root of 6x^3 + 18x^2 + 12x + 2 = 0.

Original entry on oeis.org

2, 5, 7, 7, 7, 2, 8, 0, 1, 0, 3, 1, 4, 4, 0, 8, 4, 4, 7, 2, 9, 4, 4, 9, 3, 9, 7, 2, 7, 0, 6, 3, 5, 8, 2, 2, 7, 0, 8, 9, 4, 4, 1, 2, 5, 7, 0, 0, 9, 7, 7, 3, 1, 9, 7, 8, 2, 3, 1, 4, 6, 3, 9, 3, 9, 5, 8, 0, 8, 6, 4, 4, 5, 7, 6, 7, 3, 0, 5, 3, 7, 0, 8, 5, 8, 2, 4, 9, 9, 8, 0, 0, 3, 1, 0, 1, 5, 7, 2, 3

Offset: 0

Frank M Jackson, Nov 08 2011

If the side lengths of a quadrilateral form a harmonic progression in the ratio 1 : 1/(1+d) : 1/(1+2d) : 1/(1+3d) where d is the common difference between the denominators of the harmonic progression, then the triangle inequality condition requires that d be in the range f < d < g, where f = -0.257772801... and is the greatest root of the equation: 2 + 12d + 18d^2 + 6d^3 = 0. The value of g is given in A199589.

			-0.257772801031440844729449397270635822708944125700977319782314639395808...

Index entries for algebraic numbers, degree 3

Cf. A010503, A199220, A199221, A199589.

Mathematica
```
N[Reduce[2+12d+18d^2+6d^3==0, d], 100]
```

real(polroots(6*x^3+18*x^2+12*x+2)[3]) \\ Charles R Greathouse IV, Nov 10 2011

polrootsreal(6*x^3-18*x^2+12*x-2)[1] \\ Charles R Greathouse IV, Oct 27 2023

sqrt(4/3)*sin(Pi*2/9) - 1. - Charles R Greathouse IV, Nov 10 2011

a(99) corrected by Sean A. Irvine, Jul 25 2021

cp's OEIS Frontend

A199221 Triangle read by rows: T(n,k) = (n+1-k)|s(n,n+1-k)| - 2|s(n-1,n-k)|, where s(n,k) are the signed Stirling numbers of the first kind and 1 <= k <= n.

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A199589 Decimal expansion of the greatest root of 6x^3 - 6x - 2 = 0.

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A199590 Decimal expansion (unsigned) of the greatest root of 6x^3 + 18x^2 + 12x + 2 = 0.

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cp's OEIS Frontend

A199221 Triangle read by rows: T(n,k) = (n+1-k)*|s(n,n+1-k)| - 2*|s(n-1,n-k)|, where s(n,k) are the signed Stirling numbers of the first kind and 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A199589 Decimal expansion of the greatest root of 6x^3 - 6x - 2 = 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A199590 Decimal expansion (unsigned) of the greatest root of 6x^3 + 18x^2 + 12x + 2 = 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A199221 Triangle read by rows: T(n,k) = (n+1-k)|s(n,n+1-k)| - 2|s(n-1,n-k)|, where s(n,k) are the signed Stirling numbers of the first kind and 1 <= k <= n.