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%I #80 Aug 28 2023 08:26:01
%S 1,1,0,1,1,0,1,2,2,0,1,3,6,6,0,1,4,12,24,24,0,1,5,20,60,120,120,0,1,6,
%T 30,120,360,720,720,0,1,7,42,210,840,2520,5040,5040,0,1,8,56,336,1680,
%U 6720,20160,40320,40320,0
%N Array read by ascending antidiagonals: A(n,k) the rising factorial, also known as Pochhammer symbol, for n >= 0 and k >= 0.
%C The Pochhammer function is defined P(x,n) = x*(x+1)*...*(x+n-1). By convention P(0,0) = 1.
%C From _Antti Karttunen_, Dec 19 2015: (Start)
%C Apart from the initial row of zeros, if we discard the leftmost column and divide the rest of terms A(n,k) with (n+k) [where k is now the once-decremented column index of the new, shifted position] we get the same array back. See the given recursive formula.
%C When the numbers in array are viewed in factorial base (A007623), certain repeating patterns can be discerned, at least in a few of the topmost rows. See comment in A001710 and arrays A265890, A265892. (End)
%C A(n,k) is the k-th moment (about 0) of a gamma (Erlang) distribution with shape parameter n and rate parameter 1. - _Geoffrey Critzer_, Dec 24 2018
%D Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley, 1994.
%D H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 355.
%H NIST Digital Library of Mathematical Functions, <a href="http://dlmf.nist.gov/5.2#iii">Pochhammer's Symbol</a>
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A(n,k) = Gamma(n+k)/Gamma(n) for n > 0 and n^k for n=0.
%F A(n,k) = Sum_{j=0..k} n^j*S1(k,j), S1(n,k) the Stirling cycle numbers A132393(n,k).
%F A(n,k) = (k-1)!/(Sum_{j=0..k-1} (-1)^j*binomial(k-1, j)/(j+n)) for n >= 1, k >= 1.
%F A(n,k) = (n+k-1)*A(n,k-1) for k >= 1, A(n,0) = 1. - _Antti Karttunen_, Dec 19 2015
%F E.g.f. for row k: 1/(1-x)^k. - _Geoffrey Critzer_, Dec 24 2018
%F A(n, k) = FallingFactorial(n + k - 1, k). - _Peter Luschny_, Mar 22 2022
%F G.f. for row n as a continued fraction of Stieltjes type: 1/(1 - n*x/(1 - x/(1 - (n+1)*x/(1 - 2*x/(1 - (n+2)*x/(1 - 3*x/(1 - ... ))))))). See Wall, Chapter XVIII, equation 92.5. Cf. A226513. - _Peter Bala_, Aug 27 2023
%e Square array A(n,k) [where n=row, k=column] is read by ascending antidiagonals as:
%e A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), A(3,0), A(2,1), A(1,2), A(0,3), ...
%e Array starts:
%e n\k [0 1 2 3 4 5 6 7 8]
%e --------------------------------------------------------------
%e [0] [1, 0, 0, 0, 0, 0, 0, 0, 0]
%e [1] [1, 1, 2, 6, 24, 120, 720, 5040, 40320]
%e [2] [1, 2, 6, 24, 120, 720, 5040, 40320, 362880]
%e [3] [1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400]
%e [4] [1, 4, 20, 120, 840, 6720, 60480, 604800, 6652800]
%e [5] [1, 5, 30, 210, 1680, 15120, 151200, 1663200, 19958400]
%e [6] [1, 6, 42, 336, 3024, 30240, 332640, 3991680, 51891840]
%e [7] [1, 7, 56, 504, 5040, 55440, 665280, 8648640, 121080960]
%e [8] [1, 8, 72, 720, 7920, 95040, 1235520, 17297280, 259459200]
%e .
%e Seen as a triangle, T(n, k) = Pochhammer(n - k, k), the first few rows are:
%e [0] 1;
%e [1] 1, 0;
%e [2] 1, 1, 0;
%e [3] 1, 2, 2, 0;
%e [4] 1, 3, 6, 6, 0;
%e [5] 1, 4, 12, 24, 24, 0;
%e [6] 1, 5, 20, 60, 120, 120, 0;
%e [7] 1, 6, 30, 120, 360, 720, 720, 0;
%e [8] 1, 7, 42, 210, 840, 2520, 5040, 5040, 0;
%e [9] 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320, 0.
%p for n from 0 to 8 do seq(pochhammer(n,k), k=0..8) od;
%t Table[Pochhammer[n, k], {n, 0, 8}, {k, 0, 8}]
%o (Sage)
%o for n in (0..8): print([rising_factorial(n,k) for k in (0..8)])
%o (Scheme)
%o (define (A265609 n) (A265609bi (A025581 n) (A002262 n)))
%o (define (A265609bi row col) (if (zero? col) 1 (* (+ row col -1) (A265609bi row (- col 1)))))
%o ;; _Antti Karttunen_, Dec 19 2015
%Y Triangle giving terms only up to column k=n: A124320.
%Y Row 0: A000007, row 1: A000142, row 3: A001710 (from k=1 onward, shifted two terms left).
%Y Column 0: A000012, column 1: A001477, column 2: A002378, columns 3-7: A007531, A052762, A052787, A053625, A159083 (shifted 2 .. 6 terms left respectively, i.e. without the extra initial zeros), column 8: A239035.
%Y Row sums of the triangle: A000522.
%Y A(n, n) = A000407(n-1) for n>0.
%Y 2^n*A(1/2,n) = A001147(n).
%Y Cf. also A007623, A008279 (falling factorial), A173333, A257505, A265890, A265892.
%K nonn,tabl
%O 0,8
%A _Peter Luschny_, Dec 19 2015
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