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%I #25 Sep 08 2022 08:46:11
%S 1,1,1,1,1,1,1,2,1,1,1,6,3,1,1,1,24,15,4,1,1,1,120,105,28,5,1,1,1,720,
%T 945,280,45,6,1,1,1,5040,10395,3640,585,66,7,1,1,1,40320,135135,58240,
%U 9945,1056,91,8,1,1,1,362880,2027025,1106560,208845,22176,1729,120,9,1,1
%N Table of k-fold factorials, read by antidiagonals.
%C A variant of A142589.
%H G. C. Greubel, <a href="/A256268/b256268.txt">Antidiagonal rows n = 0..100, flattened</a>
%F A(n, k) = (-n)^k*FallingFactorial(-1/n, k) for n >= 1. - _Peter Luschny_, Dec 21 2021
%e 1 1 1 1 1 1 1... A000012
%e 1 1 2 6 24 120 720... A000142
%e 1 1 3 15 105 945 10395... A001147
%e 1 1 4 28 280 3640 58240... A007559
%e 1 1 5 45 585 9945 208845... A007696
%e 1 1 6 66 1056 22176 576576... A008548
%e 1 1 7 91 1729 43225 1339975... A008542
%e 1 1 8 120 2640 76560 2756160... A045754
%e 1 1 9 153 3825 126225 5175225... A045755
%e 1 1 10 190 5320 196840 9054640... A045756
%e 1 1 11 231 7161 293601 14977651... A144773
%p seq(seq( mul(j*k+1, j=0..n-k-1), k=0..n), n=0..12); # _G. C. Greubel_, Mar 04 2020
%t T[n_, k_]= Product[j*k+1, {j,0,n-1}]; Table[T[n-k,k], {n,0,12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Mar 04 2020 *)
%o (PARI) T(n,k) = prod(j=0, n-1, j*k+1);
%o for(n=0,12, for(k=0, n, print1(T(n-k, k), ", "))) \\ _G. C. Greubel_, Mar 04 2020
%o (Magma)
%o function T(n,k)
%o if k eq 0 or n eq 0 then return 1;
%o else return (&*[j*k+1: j in [0..n-1]]);
%o end if; return T; end function;
%o [T(n-k,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 04 2020
%o (Sage) [[ product(j*k+1 for j in (0..n-k-1)) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Mar 04 2020
%o (GAP) Flat(List([0..12], n-> List([0..n], k-> Product([0..n-k-1], j-> j*k+1) ))); # _G. C. Greubel_, Mar 04 2020
%Y Cf. Columns : A000012, A000012, A000384, A011199, A011245.
%Y Cf. Diagonals : A092985, A076111, A158887.
%Y Cf. A048994, A132393.
%Y Cf. A000142 ("1-fold"), A001147 (2-fold), A007559 (3), A007696 (4), A008548 (5), A008542 (6), A045754 (7), A045755 (8), A045756 (9), A144773 (10)
%K nonn,tabl
%O 0,8
%A _Philippe Deléham_, Jun 01 2015
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