A171890 Octonomial coefficient array.

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, 42, 46, 48, 48, 46, 42, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 120, 161, 204, 246, 284, 315, 336, 344, 336, 315, 284, 246, 204, 161, 120, 84, 56, 35

Offset

0,11

Comments

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

Links

T. D. Noe, Rows n = 0..25, flattened

Formula

Row n has g.f. (1+x+...+x^7)^n.

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

Example

Array begins:
[1]
[1, 1, 1, 1, 1, 1, 1, 1]
[1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1]
...

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 8-nomials as a table
r := 8:  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

Mathematica

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 *)

Prog

(PARI) concat(vector(5, k, Vec(sum(j=0, 7, x^j)^k)))  \\ M. F. Hasler, Jun 17 2012

Crossrefs

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

Sequence in context: A325351 A363776 A279319 * A287793 A073795 A017893

Adjacent sequences: A171887 A171888 A171889 * A171891 A171892 A171893

Keyword

nonn,tabf

Author

N. J. A. Sloane, Oct 19 2010

Status

approved