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A368119
Array read by ascending antidiagonals. A(n, k) = Product_{j=0..k-1} (n*j + 1).
0
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 6, 1, 1, 1, 4, 15, 24, 1, 1, 1, 5, 28, 105, 120, 1, 1, 1, 6, 45, 280, 945, 720, 1, 1, 1, 7, 66, 585, 3640, 10395, 5040, 1, 1, 1, 8, 91, 1056, 9945, 58240, 135135, 40320, 1, 1, 1, 9, 120, 1729, 22176, 208845, 1106560, 2027025, 362880, 1
OFFSET
0,9
COMMENTS
A(n, k) is the number of increasing (n + 1)-ary trees on k vertices. (Following a comment of David Callan in A007559.)
FORMULA
Let rf(n, k) denote the rising factorial and ff(n,k) the falling factorial.
A(n, k) = n^k * rf(1/n, k) if n > 0 else 1.
A(n, k) = (-n)^k * ff(-1/n, k) if n > 0 else 1.
A(n, k) = (n^k * Gamma(k + 1/n)) / Gamma(1/n) for n > 0.
A(n, k) = ((-n)^k * Gamma(1 - 1/n)) / Gamma(1 - 1/n - k) for n > 0.
A(n, k) = k! * [x^k](1 - n*x)^(-1/n).
A(n, k) = [x^k] hypergeom([1, 1/n], [], n*x).
Column n + 1 has a linear recurrence with constant coefficients and signature ((-1)^k*binomial(n+1, n-k) for k=0..n).
EXAMPLE
Array A(n, k) starts:
[0] 1, 1, 1, 1, 1, 1, 1, 1, ... A000012
[1] 1, 1, 2, 6, 24, 120, 720, 5040, ... A000142
[2] 1, 1, 3, 15, 105, 945, 10395, 135135, ... A001147
[3] 1, 1, 4, 28, 280, 3640, 58240, 1106560, ... A007559
[4] 1, 1, 5, 45, 585, 9945, 208845, 5221125, ... A007696
[5] 1, 1, 6, 66, 1056, 22176, 576576, 17873856, ... A008548
[6] 1, 1, 7, 91, 1729, 43225, 1339975, 49579075, ... A008542
[7] 1, 1, 8, 120, 2640, 76560, 2756160, 118514880, ... A045754
[8] 1, 1, 9, 153, 3825, 126225, 5175225, 253586025, ... A045755
PROG
(SageMath)
def A(n, k): return n**k * rising_factorial(1/n, k) if n > 0 else 1
for n in range(9): print([A(n, k) for k in range(8)])
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Dec 18 2023
STATUS
approved