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%I #54 Mar 01 2020 01:10:57
%S 1,1,1,1,1,1,1,1,1,1,1,1,2,3,4,5,6,7,8,9,10,9,8,7,6,5,4,3,2,1,1,3,6,
%T 10,15,21,28,36,45,55,63,69,73,75,75,73,69,63,55,45,36,28,21,15,10,6,
%U 3,1,1,4,10,20,35,56,84,120,165,220,282,348,415,480
%N 10-nomial coefficient array: Coefficients of the polynomial (1 + ... + X^9)^n, n=0,1,...
%C The n-th row also yields the number of ways to get a total of n, n+1, ..., 10n, when throwing n 10-sided dice, or summing n integers ranging from 1 to 10.
%C The row sums equal 10^n = A011557(n).
%C The row lengths are 1 + 9n = 10n - (n-1) = A017173(n).
%C T(n,k) is the number of integers in the [0, 10^n-1] range distributed according to the sum k of their digits. - _Miquel Cerda_, Jun 21 2017
%C The sum of the squares of the integers of the n-th row gives A174061(n). - _Miquel Cerda_, Jul 03 2017
%H Seiichi Manyama, <a href="/A213651/b213651.txt">Rows n = 0..46, flattened</a>
%H Miquel Cerda, <a href="/A213651/a213651_1.pdf">Graphical construction of the triangle T(n,k) for n = 0..11.</a>
%F T(n,k) = Sum_{i = 0..floor(k/10)} (-1)^i*binomial(n,i)*binomial(n+k-1-10*i,n-1) for n >= 0 and 0 <= k <= 9*n. - _Peter Bala_, Sep 07 2013
%e There are 1, 3, 6, 10, ... ways to score a total of 4, 5, 6, 7, ... when throwing three 10-sided dice.
%e The table begins as follows:
%e (row n=0) 1; (row sum = 1, row length = 1)
%e (row n=1) 1,1,1,1,1,1,1,1,1,1; (row sum = 10, row length = 10)
%e (row n=2) 1,2,3,4,5,6,7,8,9,10,9,8,7,6,5,4,3,2,1; (sum = 100, length = 19)
%e (row n=3) 1,3,6,10,15,21,28,36,45,55,63,69,73,75,75,73,...; row sum = 1000;
%e (row n=4) 1,4,10,20,35,56,84,120,165,220,282,348,415,...; row sum = 10^4;
%e etc.
%e Number of integers in (row n=2): k(2)=3, because in the range 0 to 99 there are 3 integers whose digits sum to 2: 2, 11 and 20. - _Miquel Cerda_, Jun 21 2017
%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 10-nomials as a table
%p r := 10: 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
%o (PARI) concat(vector(5,k,Vec(sum(j=0,9,x^j)^(k-1))))
%Y The q-nomial arrays are for q=2..10: A007318 (Pascal), A027907, A008287, A035343, A063260, A063265, A171890, A213652, A213651.
%K nonn,tabf
%O 0,13
%A _M. F. Hasler_, Jun 17 2012
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