OFFSET
0,9
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.
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77,78.
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 [math.NT], 2008.
Iain G. Johnston, Optimal strategies in the Fighting Fantasy gaming system: influencing stochastic dynamics by gambling with limited resource, arXiv:2002.10172 [cs.AI], 2020.
FORMULA
G.f. for row n: (Sum_{j=0..5} x^j)^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) and with N6(6,x) = 5 - 10*x + 10*x^2 - 5*x^3 + x^4.
T(n, k) = 0 if n=-1 or k<0 or k >= 5*n + 1; T(0, 0)=1; T(n, k) = Sum_{j=0..5} T(n-1, k-j) else.
T(n, k) = Sum_{i = 0..floor(k/6)} (-1)^i*binomial(n,i)*binomial(n+k-1-6*i,n-1) for n >= 0 and 0 <= k <= 5*n. - Peter Bala, Sep 07 2013
T(n, k) = Sum_{i = max(0,ceiling((k-2*n)/3)).. min(n,k/3)} binomial(n,i)*trinomial(n,k-3*i) for n >= 0 and 0 <= k <= 5*n. - Matthew Monaghan, Sep 30 2015
EXAMPLE
The irregular table T(n, k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1: 1
2: 1 1 1 1 1 1
3: 1 2 3 4 5 6 5 4 3 2 1
4: 1 3 6 10 15 21 25 27 27 25 21 15 10 6 3 1
...reformatted - Wolfdieter Lang, Oct 31 2015
MAPLE
#Define the r-nomial coefficients for r = 1, 2, 3, ...
rnomial := (r, n, k) -> add((-1)^i*binomial(n, i)*binomial(n+k-1-r*i, n-1), i = 0..floor(k/r)):
#Display the 6-nomials as a table
r := 6: rows := 10:
for n from 0 to rows do
seq(rnomial(r, n, k), k = 0..(r-1)*n)
end do;
# Peter Bala, Sep 07 2013
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 *)
PROG
(PARI) concat(vector(5, k, Vec(sum(j=0, 5, x^j)^k))) \\ M. F. Hasler, Jun 17 2012
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
Wolfdieter Lang, Jul 24 2001
EXTENSIONS
More terms and corrected recurrence from Nicholas M. Makin (NickDMax(AT)yahoo.com), Sep 13 2002
STATUS
approved