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 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. 3
 -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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 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 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. LINKS FORMULA 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. EXAMPLE Triangle starts:   -1;    0,  1;    1,  4,   2;    2, 12,  18,   6;    3, 28,  83,  88,  24;    4, 55, 270, 575, 500, 120; MATHEMATICA Flatten[Table[(n+1-k)Abs[StirlingS1[n, n+1-k]]-2Abs[StirlingS1[n-1, n-k]], {n, 1, 20}, {k, 1, n}]] PROG (PARI) 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 CROSSREFS Cf. A094638, A192918, A199220. Sequence in context: A182700 A136202 A075418 * A096870 A261253 A328334 Adjacent sequences:  A199218 A199219 A199220 * A199222 A199223 A199224 KEYWORD sign,tabl AUTHOR Frank M Jackson, Nov 04 2011 STATUS approved

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Last modified July 10 12:34 EDT 2020. Contains 335576 sequences. (Running on oeis4.)