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A063260
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Sextinomial (also called hexanomial) coefficient array.
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14
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1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 25, 27, 27, 25, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 80, 104, 125, 140, 146, 140, 125, 104, 80, 56, 35, 20, 10, 4, 1, 1, 5, 15, 35, 70, 126, 205, 305, 420, 540, 651, 735, 780
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,9
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COMMENTS
| The sequence of step width of this staircase array is [1,5,5,...], hence the degree sequence for the row polynomials is [0,5,10,15,...]=A008587.
The column sequences (without leading zeros) are for k=0..5 those of the lower triangular array A007318 (Pascal) and for k=6..9: A062989, A063262-4. Row sums give A000400 (powers of 6). Central coefficients give A063419; see also A018901.
This can be used to calculate the number of occurrences of a given roll of n six-sided dice, where k is the index: k=0 being the lowest possible roll (i.e. n) and n*6 being the highest roll.
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77,78.
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LINKS
| T. D. Noe, Rows n = 0..25, flattened
S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle (arXiv:0802.2654)
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FORMULA
| G.f. for row n: (sum(x^j, j=0..5))^n.
G.f. for column k: (x^(ceiling(k/5)))*N6(k, x)/(1-x)^(k+1) with the row polynomials from the staircase array A063261(k, m).
a(n, k)=0 if n=-1 or k<0 or k >= 5*n + 1; a(0, 0)=1; a(n, k)= sum(a(n-1, k-j), j=0..5) else.
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EXAMPLE
| {1}; {1, 1, 1, 1, 1, 1}; {1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1}; ...
N6(k,x)= 1 for k=0..5; N6(6,x)= 5-10*x+10*x^2-5*x^3+x^4 (from A063261).
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MATHEMATICA
| Flatten[Table[CoefficientList[(1 + x + x^2 + x^3 + x^4 + x^5)^n, x], {n, 0, 25}]] (* T. D. Noe, Apr 04 2011 *)
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CROSSREFS
| The q-nomial arrays for q=2..5 are: A007318 (Pascal), A027907, A008287, A035343 and for q=7: A063265.
Sequence in context: A070667 A122416 A134665 * A073793 A017891 A017881
Adjacent sequences: A063257 A063258 A063259 * A063261 A063262 A063263
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KEYWORD
| nonn,easy,tabf
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jul 24 2001
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EXTENSIONS
| More terms and corrected recurrence from Nicholas M. Makin (NickDMax(AT)yahoo.com), Sep 13 2002
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