OFFSET
1,2
COMMENTS
The array is trapezoidal rather than triangular because C(n,k) is not uniquely defined for all negative n and negative k.
Row sums are 0.
Coefficients of (x-1)^7 (x+1)^(n-1), n > 0.
LINKS
G. C. Greubel, Rows n=1..50 of trapezoid, flattened
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
FORMULA
T(n,k) = Sum_{i=0..n+m-1} (-1)^(i+m)*C(m,i)*C(n-1,k-i), n >= 1, with T(n,0) = (-1)^m and m=7.
EXAMPLE
Trapezoid begins:
-1, 7, -21, 35, -35, 21, -7, 1;
-1, 6, -14, 14, 0, -14, 14, -6, 1;
-1, 5, -8, 0, 14, -14, 0, 8, -5, 1;
-1, 4, -3, -8, 14, 0, -14, 8, 3, -4, 1;
-1, 3, 1, -11, 6, 14, -14, -6, 11, -1, -3, 1;
-1, 2, 4, -10, -5, 20, 0, -20, 5, 10, -4, -2, 1;
-1, 1, 6, -6, -15, 15, 20, -20, -15, 15, 6, -6, -1, 1;
etc.
MATHEMATICA
Flatten[Table[CoefficientList[(x - 1)^7 (x + 1)^n, x], {n, 0, 7}]] (* T. D. Noe, Oct 25 2013 *)
m=7; Table[If[k == 0, (-1)^m, Sum[(-1)^(j+m)*Binomial[m, j]*Binomial[n-1, k-j], {j, 0, n+m-1}]], {n, 1, 10}, {k, 0, n+m-1}]//Flatten (* G. C. Greubel, Nov 28 2018 *)
PROG
(PARI) m=7; for(n=1, 10, for(k=0, n+m-1, print1(if(k==0, (-1)^m, sum(j=0, n+m-1, (-1)^(j+m)*binomial(m, j)*binomial(n-1, k-j))), ", "))) \\ G. C. Greubel, Nov 28 2018
(Magma) m:=7; [[k le 0 select (-1 )^m else (&+[(-1)^(j+m)* Binomial(m, j) *Binomial(n-1, k-j): j in [0..(n+m-1)]]): k in [0..(n+m-1)]]: n in [1..10]]; // G. C. Greubel, Nov 28 2018
(Sage) m=7; [[sum((-1)^(j+m)*binomial(m, j)*binomial(n-1, k-j) for j in range(n+m)) for k in range(n+m)] for n in (1..10)] # G. C. Greubel, Nov 28 2018
CROSSREFS
KEYWORD
easy,sign,tabf
AUTHOR
Dixon J. Jones, Oct 12 2013
STATUS
approved