A137267 Chung-Graham juggling polynomials as a triangular sequence of positive coefficients.

1, 1, 1, 2, 2, 2, 3, 3, 4, 6, 4, 4, 6, 12, 24, 5, 5, 8, 18, 48, 120, 6, 6, 10, 24, 72, 240, 720, 7, 7, 12, 30, 96, 360, 1440, 5040, 8, 8, 14, 36, 120, 480, 2160, 10080, 40320, 9, 9, 16, 42, 144, 600, 2880, 15120, 80640, 362880

Offset: 1

Author

Roger L. Bagula, Mar 12 2008

Keywords

• nonn
• tabl

Comments

Row sums are {1, 2, 6, 16, 50, 204, 1078, 6992, 53226, 462340,..} which is A014144(n) - 1 for n>=2.

Row sums are given by (n+1)* !n - !(n+1), for n>=2, where !n is the left factorial (A003422). - G. C. Greubel, Mar 27 2019

Examples

			Triangle begins with:
1;
1, 1;
2, 2,  2;
3, 3,  4,  6;
4, 4,  6, 12,  24;
5, 5,  8, 18,  48, 120;
6, 6, 10, 24,  72, 240,  720;
7, 7, 12, 30,  96, 360, 1440,  5040;
8, 8, 14, 36, 120, 480, 2160, 10080, 40320;
9, 9, 16, 42, 144, 600, 2880, 15120, 80640, 362880;
		

Links

• G. C. Greubel, Rows n=1..100 of triangle, flattened
• Fan Chung, R. L. Graham, Primitive juggling sequences, Am. Math. Monthly 115 (3) (2008) 185-194

Crossrefs

Cf. A137948.

Programs

Mathematica

p[x_, n_]:= If[n == 1, 1, -(1 - Sum[(n-k)*k!*x^k, {k, 0, n-1}])]; Table[CoefficientList[p[x, n], x], {n, 1, 10}]//Flatten

Formula

Given f_b(x) = (1 - Sum_{k=0..n-1} (n-k)*k!*x^k)/(1-(b+1)*x), then

p(x,b) = -f_b(x)*(1-(b+1)*x) = -(1 - Sum_{k=0..n-1} (n-k)*k!*x^k ).