A199220 Triangle read by rows: T(n,k) = (n-1-k)*abs(s(n,n+1-k)), where s(n,k) are the signed Stirling numbers of the first kind and 1 <= k <= n.

-1, 0, -1, 1, 0, -2, 2, 6, 0, -6, 3, 20, 35, 0, -24, 4, 45, 170, 225, 0, -120, 5, 84, 525, 1470, 1624, 0, -720, 6, 140, 1288, 5880, 13538, 13132, 0, -5040, 7, 216, 2730, 18144, 67347, 134568, 118124, 0, -40320, 8, 315, 5220, 47250, 253092, 807975, 1447360, 1172700, 0, -362880, 9, 440, 9240, 108900, 788865, 3608220, 10250790, 16819000, 12753576, 0, -3628800

Offset: 1

Author

Frank M Jackson, Nov 04 2011

Keywords

• sign
• tabl

Comments

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 g(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 lower limit f(n) is the greatest root of another group of polynomials defined by coefficients in the triangle A199221.

Examples

			Triangle starts:
  -1;
   0, -1;
   1,  0,  -2;
   2,  6,   0,  -6;
   3, 20,  35,   0, -24;
   4, 45, 170, 225,   0, -120;
		

Crossrefs

Cf. A094638, A192918, A199221.

Programs

Mathematica

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

PARI

T(n,k) = (n-1-k)*abs(stirling(n,n+1-k,1)); \\ Michel Marcus, Sep 30 2018

Formula

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