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%I #21 Feb 29 2020 13:35:07
%S 1,1,1,1,1,1,1,1,1,1,1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1,1,3,6,10,15,21,
%T 28,36,45,52,57,60,61,60,57,52,45,36,28,21,15,10,6,3,1,1,4,10,20,35,
%U 56,84,120,165,216,270,324,375,420,456,480,489,480,456
%N 9-nomial coefficient array: Coefficients of the polynomial (1+...+X^8)^n, n=0,1,...
%C The n-th row also yields the number of ways to get a total of n, n+1,..., 9n, when summing n integers ranging from 1 to 9.
%C The row sums equal 9^n = A001019(n).
%C The row lengths are 1+8n = A017077(n).
%H Seiichi Manyama, <a href="/A213652/b213652.txt">Rows n = 0..49, flattened</a>
%F T(n,k) = Sum_{i=0..floor(k/9)} (-1)^i*binomial(n,i)*binomial(n+k-1-9*i,n-1) for n >= 0 and 0 <= k <= 8*n. - _Peter Bala_, Sep 07 2013
%e The triangle starts:
%e (row n=0) 1; (row sum = 1, row length = 1)
%e (row n=1) 1,1,1,1,1,1,1,1,1; (row sum = 9, row length = 9)
%e (row n=2) 1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1; (sum = 81, length = 17)
%e (row n=3) 1,3,6,10,15,21,28,36,45,52,57,60,61,60,... (sum = 729, length = 25)
%e (row n=4) 1, 4, 10, 20, 35, 56, 84, 120, 165, 216, 270, 324, 375, 420, 456,... (sum = 9^4; length = 33),
%e etc.
%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 9-nomials as a table
%p r := 9: rows := 10:
%p for n from 0 to rows do
%p seq(rnomial(r,n,k), k = 0..(r-1)*n)
%p end do; # _Peter Bala_, Sep 07 2013
%o (PARI) concat(vector(5,k,Vec(sum(j=0,8,x^j)^(k-1))))
%Y The q-nomial arrays are for q=2..10: A007318 (Pascal), A027907, A008287, A035343, A063260, A063265, A171890, A213652, A213651.
%Y Cf. A001019, A017077.
%K nonn,tabf
%O 0,12
%A _M. F. Hasler_, Jun 17 2012
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