A063265 Septinomial (also called heptanomial) coefficient array.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 33, 36, 37, 36, 33, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 116, 149, 180, 206, 224, 231, 224, 206, 180, 149, 116, 84, 56, 35

Offset

0,10

Comments

The sequence of step width of this staircase array is [1,6,6,...], hence the degree sequence for the row polynomials is [0,6,12,18,...]= A008588.

The column sequences (without leading zeros) are for k=0..6 those of the lower triangular array A007318 (Pascal) and for k=7..9: A063267, A063417, A063418. Row sums give A000420 (powers of 7). Central coefficients give A025012.

References

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77,78.

Links

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

S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle (arXiv:0802.2654)

Formula

a(n, k)=0 if n=-1 or k<0 or k >= 6*n; a(0, 0)=1; a(n, k)= sum(a(n-1, k-j), j=0..6) else.

G.f. for row n: (sum(x^j, j=0..6))^n.

G.f. for column k: (x^(ceiling(k/6)))*N7(k, x)/(1-x)^(k+1) with the row polynomials of the staircase array A063266(k, m).

T(n,k) = sum {i = 0..floor(k/7)} (-1)^i*binomial(n,i)*binomial(n+k-1-7*i,n-1) for n >= 0 and 0 <= k <= 6*n. - Peter Bala, Sep 07 2013

Example

{1};
{1, 1, 1, 1, 1, 1, 1};
{1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1};
...
N7(k,x)= 1 for k=0..6, N7(7,x)= 6-15*x+20*x^2-15*x^3+6*x^4-x^5 (from A063266).

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 7-nomials as a table
r := 7:  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)^n, x], {n, 0, 25}]] (* T. D. Noe, Apr 04 2011 *)

Crossrefs

The q-nomial arrays are for q=2..8: A007318 (Pascal), A027907, A008287, A035343, A063260, A063265, A171890.

Sequence in context: A307785 A331305 A279313 * A211011 A232242 A287655

Adjacent sequences: A063262 A063263 A063264 * A063266 A063267 A063268

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang, Jul 24 2001

Status

approved