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 A063260 Sextinomial (also called hexanomial) coefficient array. 20
 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; text; internal format)
 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. 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. a(n, k)=0 if n=-1 or k<0 or k >= 5*n + 1; a(0, 0)=1; a(n, k)= Sum_{j=0..5} a(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 The q-nomial arrays for q=2..5 are: A007318 (Pascal), A027907, A008287, A035343 and for q=7: A063265, A171890, A213652, A213651. Sequence in context: A122416 A134665 A271832 * A271859 A232240 A073793 Adjacent sequences:  A063257 A063258 A063259 * A063261 A063262 A063263 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

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Last modified November 21 12:34 EST 2017. Contains 295001 sequences.