

A137267


ChungGraham juggling polynomials as a triangular sequence of positive coefficients.


2



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
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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) 185194


FORMULA

Given f_b(x) = (1  Sum_{k=0..n1} (nk)*k!*x^k)/(1(b+1)*x), then
p(x,b) = f_b(x)*(1(b+1)*x) = (1  Sum_{k=0..n1} (nk)*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[(nk)*k!*x^k, {k, 0, n1}])]; Table[CoefficientList[p[x, n], x], {n, 1, 10}]//Flatten


CROSSREFS

Cf. A137948.
Sequence in context: A074732 A089046 A054911 * A341451 A123576 A094824
Adjacent sequences: A137264 A137265 A137266 * A137268 A137269 A137270


KEYWORD

nonn,tabl


AUTHOR

Roger L. Bagula, Mar 12 2008


STATUS

approved



