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 A216919 The Gauss factorial N_n! for N >= 0, n >= 1, square array read by antidiagonals. 11
 1, 1, 1, 2, 1, 1, 6, 1, 1, 1, 24, 3, 2, 1, 1, 120, 3, 2, 1, 1, 1, 720, 15, 8, 3, 2, 1, 1, 5040, 15, 40, 3, 6, 1, 1, 1, 40320, 105, 40, 15, 24, 1, 2, 1, 1, 362880, 105, 280, 15, 24, 1, 6, 1, 1, 1, 3628800, 945, 2240, 105, 144, 5, 24, 3, 2, 1, 1, 39916800, 945 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS The term is due to Cosgrave & Dilcher. The Gauss factorial should not be confused with the q-factorial [n]_q! which is also called Gaussian factorial. LINKS Alois P. Heinz, Antidiagonals n = 1..141, flattened J. B. Cosgrave, K. Dilcher, Extensions of the Gauss-Wilson Theorem, Integers: Electronic Journal of Combinatorial Number Theory, 8 (2008). J. B. Cosgrave, K. Dilcher, An Introduction to Gauss Factorials, The American Mathematical Monthly, Vol. 118, No. 9 (2011), 812-829. K. Dilcher, Gauss Factorials: Properties and Applications. Video by the Irmacs Centre, May 18, 2011. FORMULA N_n! = product_{1<=j<=N, GCD(j,n)=1} j. EXAMPLE [n\N][0, 1, 2, 3,  4,   5,   6,    7,     8,      9,     10] ------------------------------------------------------------ [  1] 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 [A000142] [  2] 1, 1, 1, 3,  3,  15,  15,  105,   105,    945,     945 [A055634, A133221] [  3] 1, 1, 2, 2,  8,  40,  40,  280,  2240,   2240,   22400 [A232980] [  4] 1, 1, 1, 3,  3,  15,  15,  105,   105,    945,     945 [  5] 1, 1, 2, 6, 24,  24, 144, 1008,  8064,  72576,   72576 [A232981] [  6] 1, 1, 1, 1,  1,   5,   5,   35,    35,     35,      35 [A232982] [  7] 1, 1, 2, 6, 24, 120, 720,  720,  5760,  51840,  518400 [A232983] [  8] 1, 1, 1, 3,  3,  15,  15,  105,   105,    945,     945 [  9] 1, 1, 2, 2,  8,  40,  40,  280,  2240,   2240,   22400 [ 10] 1, 1, 1, 3,  3,   3,   3,   21,    21,    189,     189 [A232984] [ 11] 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 [A232985] [ 12] 1, 1, 1, 1,  1,   5,   5,   35,    35,     35,      35 [ 13] 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 MAPLE A:= (n, N)-> mul(`if`(igcd(j, n)=1, j, 1), j=1..N): seq (seq (A(n, d-n), n=1..d), d=1..12);  # Alois P. Heinz, Oct 03 2012 MATHEMATICA GaussFactorial[m_, n_] := Product[ If[ GCD[j, n] == 1, j, 1], {j, 1, m}]; Table[ GaussFactorial[m - n, n], {m, 1, 12}, {n, 1, m}] // Flatten (* Jean-François Alcover, Mar 18 2013 *) PROG (Sage) def Gauss_factorial(N, n): return mul(j for j in (1..N) if gcd(j, n) == 1) for n in (1..13): [Gauss_factorial(N, n) for N in (0..10)] (PARI) T(m, n)=prod(k=2, m, if(gcd(k, n)==1, k, 1)) for(s=1, 10, for(n=1, s, print1(T(s-n, n)", "))) \\ Charles R Greathouse IV, Oct 01 2012 CROSSREFS A000142(n) = n! = Gauss_factorial(n, 1). A001147(n) = Gauss_factorial(2*n, 2). A055634(n) = Gauss_factorial(n, 2)*(-1)^n. A001783(n) = Gauss_factorial(n, n). A124441(n) = Gauss_factorial(floor(n/2), n). A124442(n) = Gauss_factorial(n, n)/Gauss_factorial(floor(n/2), n). A066570(n) = Gauss_factorial(n, 1)/Gauss_factorial(n, n). Cf. A133221, A232980, A232981, A232982, A232983, A232984, A232985. Sequence in context: A204168 A216914 A216917 * A152656 A096162 A333144 Adjacent sequences:  A216916 A216917 A216918 * A216920 A216921 A216922 KEYWORD nonn,tabl AUTHOR Peter Luschny, Oct 01 2012 STATUS approved

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Last modified June 4 17:14 EDT 2020. Contains 334828 sequences. (Running on oeis4.)