A213652 9-nomial coefficient array: Coefficients of the polynomial (1+...+X^8)^n, n=0,1,...

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, 28, 36, 45, 52, 57, 60, 61, 60, 57, 52, 45, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 120, 165, 216, 270, 324, 375, 420, 456, 480, 489, 480, 456

Offset

0,12

Comments

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.

The row sums equal 9^n = A001019(n).

The row lengths are 1+8n = A017077(n).

Links

Seiichi Manyama, Rows n = 0..49, flattened

Formula

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

Example

The triangle starts:
(row n=0) 1; (row sum = 1, row length = 1)
(row n=1) 1,1,1,1,1,1,1,1,1; (row sum = 9, row length = 9)
(row n=2) 1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1; (sum = 81, length = 17)
(row n=3) 1,3,6,10,15,21,28,36,45,52,57,60,61,60,... (sum = 729, length = 25)
(row n=4) 1, 4, 10, 20, 35, 56, 84, 120, 165, 216, 270, 324, 375, 420, 456,... (sum = 9^4; length = 33),
etc.

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 9-nomials as a table
r := 9:  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

Prog

(PARI) concat(vector(5, k, Vec(sum(j=0, 8, x^j)^(k-1))))

Crossrefs

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

Cf. A001019, A017077.

Sequence in context: A081598 A232360 A158289 * A262734 A287794 A179987

Adjacent sequences: A213649 A213650 A213651 * A213653 A213654 A213655

Keyword

nonn,tabf

Author

M. F. Hasler, Jun 17 2012

Status

approved