|
%I #20 Jan 26 2021 10:27:50
%S 1,1,1,1,1,1,1,1,1,1,2,3,4,5,6,7,8,7,6,5,4,3,2,1,1,3,6,10,15,21,28,36,
%T 42,46,48,48,46,42,36,28,21,15,10,6,3,1,1,4,10,20,35,56,84,120,161,
%U 204,246,284,315,336,344,336,315,284,246,204,161,120,84,56,35
%N Octonomial coefficient array.
%C Row lengths are 1,8,15,22,... = 1+7n = A016993(n). Row sums are 1,8,64,... = 8^n = A001018(n). _M. F. Hasler_, Jun 17 2012
%H T. D. Noe, <a href="/A171890/b171890.txt">Rows n = 0..25, flattened</a>
%F Row n has g.f. (1+x+...+x^7)^n.
%F T(n,k) = sum {i = 0..floor(k/8)} (-1)^i*binomial(n,i)*binomial(n+k-1-8*i,n-1) for n >= 0 and 0 <= k <= 7*n. - _Peter Bala_, Sep 07 2013
%e Array begins:
%e [1]
%e [1, 1, 1, 1, 1, 1, 1, 1]
%e [1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1]
%e ...
%p #Define the r-nomial coefficients for r = 1, 2, 3, ...
%p rnomial := (r,n,k) -> add((-1)^i*binomial(n,i)*binomial(n+k-1-r*i,n-1), i = 0..floor(k/r)):
%p #Display the 8-nomials as a table
%p r := 8: rows := 10:
%p for n from 0 to rows do
%p seq(rnomial(r,n,k), k = 0..(r-1)*n)
%p end do;
%p # _Peter Bala_, Sep 07 2013
%t Flatten[Table[CoefficientList[(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7)^n, x], {n, 0, 10}]] (* _T. D. Noe_, Apr 04 2011 *)
%o (PARI) concat(vector(5, k, Vec(sum(j=0, 7, x^j)^k))) \\ _M. F. Hasler_, Jun 17 2012
%Y The q-nomial arrays are for q=2..10: A007318 (Pascal), A027907, A008287,A035343, A063260, A063265, A171890, A213652, A213651.
%K nonn,tabf
%O 0,11
%A _N. J. A. Sloane_, Oct 19 2010
|
