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

%I A137267 #12 Mar 27 2019 18:58:07
%S A137267 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,
%T A137267 7,7,12,30,96,360,1440,5040,8,8,14,36,120,480,2160,10080,40320,9,9,16,
%U A137267 42,144,600,2880,15120,80640,362880
%N A137267 Chung-Graham juggling polynomials as a triangular sequence of positive coefficients.
%C A137267 Row sums are {1, 2, 6, 16, 50, 204, 1078, 6992, 53226, 462340,..} which is A014144(n) - 1 for n>=2.
%C A137267 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
%H A137267 G. C. Greubel, <a href="/A137267/b137267.txt">Rows n=1..100 of triangle, flattened</a>
%H A137267 Fan Chung, R. L. Graham, <a href="http://www.jstor.org/stable/27642443">Primitive juggling sequences</a>, Am. Math. Monthly 115 (3) (2008) 185-194
%F A137267 Given f_b(x) = (1 - Sum_{k=0..n-1} (n-k)*k!*x^k)/(1-(b+1)*x), then
%F A137267 p(x,b) = -f_b(x)*(1-(b+1)*x) = -(1 - Sum_{k=0..n-1} (n-k)*k!*x^k ).
%e A137267 Triangle begins with:
%e A137267 1;
%e A137267 1, 1;
%e A137267 2, 2,  2;
%e A137267 3, 3,  4,  6;
%e A137267 4, 4,  6, 12,  24;
%e A137267 5, 5,  8, 18,  48, 120;
%e A137267 6, 6, 10, 24,  72, 240,  720;
%e A137267 7, 7, 12, 30,  96, 360, 1440,  5040;
%e A137267 8, 8, 14, 36, 120, 480, 2160, 10080, 40320;
%e A137267 9, 9, 16, 42, 144, 600, 2880, 15120, 80640, 362880;
%t A137267 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
%Y A137267 Cf. A137948.
%K A137267 nonn,tabl
%O A137267 1,4
%A A137267 _Roger L. Bagula_, Mar 12 2008