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

%I A368119 #17 Dec 18 2023 18:13:32
%S A368119 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,
%T A368119 45,280,945,720,1,1,1,7,66,585,3640,10395,5040,1,1,1,8,91,1056,9945,
%U A368119 58240,135135,40320,1,1,1,9,120,1729,22176,208845,1106560,2027025,362880,1
%N A368119 Array read by ascending antidiagonals. A(n, k) = Product_{j=0..k-1} (n*j + 1).
%C A368119 A(n, k) is the number of increasing (n + 1)-ary trees on k vertices. (Following a comment of _David Callan_ in A007559.)
%H A368119 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%F A368119 Let rf(n, k) denote the rising factorial and ff(n,k) the falling factorial.
%F A368119 A(n, k) = n^k * rf(1/n, k) if n > 0 else 1.
%F A368119 A(n, k) = (-n)^k * ff(-1/n, k) if n > 0 else 1.
%F A368119 A(n, k) = (n^k * Gamma(k + 1/n)) / Gamma(1/n) for n > 0.
%F A368119 A(n, k) = ((-n)^k * Gamma(1 - 1/n)) / Gamma(1 - 1/n - k) for n > 0.
%F A368119 A(n, k) = k! * [x^k](1 - n*x)^(-1/n).
%F A368119 A(n, k) = [x^k] hypergeom([1, 1/n], [], n*x).
%F A368119 Column n + 1 has a linear recurrence with constant coefficients and signature ((-1)^k*binomial(n+1, n-k) for k=0..n).
%e A368119 Array A(n, k) starts:
%e A368119   [0] 1, 1, 1,   1,    1,      1,       1,         1, ...  A000012
%e A368119   [1] 1, 1, 2,   6,   24,    120,     720,      5040, ...  A000142
%e A368119   [2] 1, 1, 3,  15,  105,    945,   10395,    135135, ...  A001147
%e A368119   [3] 1, 1, 4,  28,  280,   3640,   58240,   1106560, ...  A007559
%e A368119   [4] 1, 1, 5,  45,  585,   9945,  208845,   5221125, ...  A007696
%e A368119   [5] 1, 1, 6,  66, 1056,  22176,  576576,  17873856, ...  A008548
%e A368119   [6] 1, 1, 7,  91, 1729,  43225, 1339975,  49579075, ...  A008542
%e A368119   [7] 1, 1, 8, 120, 2640,  76560, 2756160, 118514880, ...  A045754
%e A368119   [8] 1, 1, 9, 153, 3825, 126225, 5175225, 253586025, ...  A045755
%o A368119 (SageMath)
%o A368119 def A(n, k): return n**k * rising_factorial(1/n, k) if n > 0 else 1
%o A368119 for n in range(9): print([A(n, k) for k in range(8)])
%Y A368119 Rows: A000012, A000142, A001147, A007559, A007696, A008548, A008542, A045754, A045755.
%Y A368119 Columns: A000012, A000027, A000384, A011199, A011245.
%Y A368119 Variant: A256268. Main diagonal: A092985.
%Y A368119 Cf. A124320, A008279, A326323.
%K A368119 nonn,tabl
%O A368119 0,9
%A A368119 _Peter Luschny_, Dec 18 2023