OFFSET
1,1
COMMENTS
n-th row sum = 1 for n >= 2.
FORMULA
T(n,k) = (-1)^(k-1) * (C(n,k) + C(n-1,k-1)), for n >= 1, k >= 1.
T(n,k) = (-1)^(k-1) * C(n,k)*(n+k)/n, for n >= 1, k >= 1.
EXAMPLE
First 7 rows:
2
3 -2
4 -5 2
5 -9 7 -2
6 -14 16 -9 2
7 -20 30 -25 11 -2
8 -27 50 -55 36 -13 2
Row 4 gives recurrence coefficients for the sequence
(r(k)) = (A002662)) = (0,0,0,1,5,16,42,99,219,...); i.e.,
r(k) = 5*r(k-1) - 9*r(k-2) + 7*r(k-3) - 2*r(k-4),
with initial values (r(0), r(1), r(2), r(3)) = (0,0,0,1).
(Here r(k) = number of subsets of {1,2,...,4} having at least 3 elements.)
MATHEMATICA
Table[Binomial[n, k]*(-1)^(k - 1)*(n + k)/n, {n, 1, 12}, {k, 1, n}]
CROSSREFS
KEYWORD
tabl,sign
AUTHOR
Clark Kimberling, Sep 24 2022
STATUS
approved