A213651 10-nomial coefficient array: Coefficients of the polynomial (1 + ... + X^9)^n, n=0,1,...

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, 10, 15, 21, 28, 36, 45, 55, 63, 69, 73, 75, 75, 73, 69, 63, 55, 45, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 282, 348, 415, 480

Offset: 0

Author

M. F. Hasler, Jun 17 2012

Keywords

• nonn
• tabf

Comments

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.

The row sums equal 10^n = A011557(n).

The row lengths are 1 + 9n = 10n - (n-1) = A017173(n).

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

The sum of the squares of the integers of the n-th row gives A174061(n). - Miquel Cerda, Jul 03 2017

Examples

			There are 1, 3, 6, 10, ... ways to score a total of 4, 5, 6, 7, ... when throwing three 10-sided dice.
The table begins as follows:
(row n=0) 1; (row sum = 1, row length = 1)
(row n=1) 1,1,1,1,1,1,1,1,1,1; (row sum = 10, row length = 10)
(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)
(row n=3) 1,3,6,10,15,21,28,36,45,55,63,69,73,75,75,73,...; row sum = 1000;
(row n=4) 1,4,10,20,35,56,84,120,165,220,282,348,415,...; row sum = 10^4;
etc.
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
		

Links

• Seiichi Manyama, Rows n = 0..46, flattened
• Miquel Cerda, Graphical construction of the triangle T(n,k) for n = 0..11.

Crossrefs

The q-nomial arrays are for q=2..10: A007318 (Pascal), A027907, A008287, A035343, A063260, A063265, A171890, A213652, A213651.

Programs

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 10-nomials as a table
r := 10:  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

PARI

concat(vector(5,k,Vec(sum(j=0,9,x^j)^(k-1))))

Formula

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