0,9

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.

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77,78.

S. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle (arXiv:0802.2654)

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.

{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).

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 this_sequence A073793 A017891 A017881

Adjacent sequences: A063257 A063258 A063259 this_sequence A063261 A063262 A063263

nonn,easy,tabf

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jul 24 2001

More terms and corrected recurrence from Nicholas M. Makin (NickDMax(AT)yahoo.com), Sep 13 2002