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A256268 Table of k-fold factorials, read by antidiagonals.

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