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A137267 Chung-Graham 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 (list; table; graph; refs; listen; history; text; internal format)
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: 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

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Last modified November 27 19:48 EST 2022. Contains 358406 sequences. (Running on oeis4.)