A368119 Array read by ascending antidiagonals. A(n, k) = Product_{j=0..k-1} (n*j + 1).

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

Author

Peter Luschny, Dec 18 2023

Keywords

• nonn
• tabl

Comments

A(n, k) is the number of increasing (n + 1)-ary trees on k vertices. (Following a comment of David Callan in A007559.)

Examples

			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
		

Links

• Index entries for sequences related to factorial numbers.

Crossrefs

Rows: A000012, A000142, A001147, A007559, A007696, A008548, A008542, A045754, A045755.

Columns: A000012, A000027, A000384, A011199, A011245.

Variant: A256268. Main diagonal: A092985.

Cf. A124320, A008279, A326323.

Programs

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)])

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).