|
%I
%S 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,
%T 10,6,3,1,1,4,10,20,35,56,80,104,125,140,146,140,125,104,80,56,35,20,
%U 10,4,1,1,5,15,35,70,126,205,305,420,540,651,735,780
%N Sextinomial (also called hexanomial) coefficient array.
%C 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.
%C 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.
%C 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.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77,78.
%H T. D. Noe, <a href="/A063260/b063260.txt">Rows n = 0..25, flattened</a>
%H S. R. Finch, P. Sebah and Z.-Q. Bai, <a href="http://arXiv.org/abs/0802.2654">Odd Entries in Pascal's Trinomial Triangle</a> (arXiv:0802.2654)
%F G.f. for row n: (sum(x^j, j=0..5))^n.
%F 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).
%F 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.
%e {1};
%e {1, 1, 1, 1, 1, 1};
%e {1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1}; ...
%e N6(k,x)= 1 for k=0..5; N6(6,x)= 5-10*x+10*x^2-5*x^3+x^4 (from A063261).
%t Flatten[Table[CoefficientList[(1 + x + x^2 + x^3 + x^4 + x^5)^n, x], {n, 0, 25}]] (* T. D. Noe, Apr 04 2011 *)
%o (PARI) concat(vector(5,k,Vec(sum(j=0,5,x^j)^k))) \\ - _M. F. Hasler_, Jun 17 2012
%Y The q-nomial arrays for q=2..5 are: A007318 (Pascal), A027907, A008287, A035343 and for q=7: A063265, A171890, A213652, A213651.
%K nonn,easy,tabf
%O 0,9
%A _Wolfdieter Lang_, Jul 24 2001
%E More terms and corrected recurrence from Nicholas M. Makin (NickDMax(AT)yahoo.com), Sep 13 2002
|
