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A292973
Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (-1)^(k+1) * k! * Sum_{i=0..n-1} (-1)^i * binomial(n-1,i) * binomial(i+1,k) * T(n-1-i,k) for n > 0.
6
1, 1, -1, 1, 1, 2, 1, 0, -1, -5, 1, 0, 2, -2, 15, 1, 0, 0, -6, 9, -52, 1, 0, 0, 6, 24, -4, 203, 1, 0, 0, 0, -24, -140, -95, -877, 1, 0, 0, 0, 24, 60, 870, 414, 4140, 1, 0, 0, 0, 0, -120, 240, -5922, 49, -21147, 1, 0, 0, 0, 0, 120, 360, -4830, 45416, -10088, 115975
OFFSET
0,6
LINKS
FORMULA
T(n,k) = n! * Sum_{j=0..floor(n/k)} (-j)^(n-k*j)/(j! * (n-k*j)!) for k > 0. - Seiichi Manyama, Jul 10 2022
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, ...
-1, 1, 0, 0, 0, ...
2, -1, 2, 0, 0, ...
-5, -2, -6, 6, 0, ...
15, 9, 24, -24, 24, ...
PROG
(Ruby)
def f(n)
return 1 if n < 2
(1..n).inject(:*)
end
def ncr(n, r)
return 1 if r == 0
(n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
ary = [1]
(1..n).each{|i| ary << (-1) ** (k % 2 + 1) * f(k) * (0..i - 1).inject(0){|s, j| s + (-1) ** (j % 2) * ncr(i - 1, j) * ncr(j + 1, k) * ary[i - 1 - j]}}
ary
end
def A292973(n)
a = []
(0..n).each{|i| a << A(i, n - i)}
ary = []
(0..n).each{|i|
(0..i).each{|j|
ary << a[i - j][j]
}
}
ary
end
p A292973(20)
CROSSREFS
Columns k=0-5 give: A292935, A003725, A292907, A292908, A292969, A292970.
Rows n=0 gives A000012.
Main diagonal gives A000142.
Sequence in context: A292948 A210872 A360753 * A220235 A066603 A263339
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, Sep 27 2017
STATUS
approved