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

1,4

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

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

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 ).

Example

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;

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

Crossrefs

Cf. A137948.

Sequence in context: A089046 A054911 A367215 * A341451 A123576 A094824

Adjacent sequences: A137264 A137265 A137266 * A137268 A137269 A137270

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 12 2008

Status

approved