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A063260 Sextinomial (also called hexanomial) coefficient array.

%I A063260 #56 May 02 2025 10:28:39
%S A063260 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 A063260 10,6,3,1,1,4,10,20,35,56,80,104,125,140,146,140,125,104,80,56,35,20,
%U A063260 10,4,1,1,5,15,35,70,126,205,305,420,540,651,735,780
%N A063260 Sextinomial (also called hexanomial) coefficient array.
%C A063260 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 A063260 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 A063260 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 A063260 L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77,78.
%H A063260 T. D. Noe, <a href="/A063260/b063260.txt">Rows n = 0..25, flattened</a>
%H A063260 Steven 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 [math.NT], 2008.
%H A063260 Iain G. Johnston, <a href="https://arxiv.org/abs/2002.10172">Optimal strategies in the Fighting Fantasy gaming system: influencing stochastic dynamics by gambling with limited resource</a>, arXiv:2002.10172 [cs.AI], 2020.
%H A063260 Yassine Otmani, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Otmani/otmani10.html">The 2-Pascal Triangle and a Related Riordan Array</a>, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 4.
%F A063260 G.f. for row n: (Sum_{j=0..5} x^j)^n.
%F A063260 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.
%F A063260 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.
%F A063260 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
%F A063260 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
%e A063260 The irregular table T(n, k) begins:
%e A063260 n\k 0 1 2  3  4  5  6  7  8  9 10 11 12 13 14 15
%e A063260 1:  1
%e A063260 2:  1 1 1  1  1  1
%e A063260 3:  1 2 3  4  5  6  5  4  3  2  1
%e A063260 4:  1 3 6 10 15 21 25 27 27 25 21 15 10  6  3  1
%e A063260 ...reformatted - _Wolfdieter Lang_, Oct 31 2015
%p A063260 #Define the r-nomial coefficients for r = 1, 2, 3, ...
%p A063260 rnomial := (r,n,k) -> add((-1)^i*binomial(n,i)*binomial(n+k-1-r*i,n-1), i = 0..floor(k/r)):
%p A063260 #Display the 6-nomials as a table
%p A063260 r := 6:  rows := 10:
%p A063260 for n from 0 to rows do
%p A063260 seq(rnomial(r,n,k), k = 0..(r-1)*n)
%p A063260 end do;
%p A063260 # _Peter Bala_, Sep 07 2013
%t A063260 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 A063260 (PARI) concat(vector(5,k,Vec(sum(j=0,5,x^j)^k)))  \\ _M. F. Hasler_, Jun 17 2012
%Y A063260 The q-nomial arrays for q=2..5 are: A007318 (Pascal), A027907, A008287, A035343 and for q=7: A063265, A171890, A213652, A213651.
%Y A063260 Columns for k=0..9 (with some shifts) are: A000012, A000027, A000217, A000292, A000332, A000389, A062989, A063262, A063263, A063264.
%Y A063260 Cf. A000400, A008587, A018901, A063261, A063419.
%K A063260 nonn,easy,tabf
%O A063260 0,9
%A A063260 _Wolfdieter Lang_, Jul 24 2001
%E A063260 More terms and corrected recurrence from Nicholas M. Makin (NickDMax(AT)yahoo.com), Sep 13 2002