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URL: https://oeis.org/A256268

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A256268
Table of k-fold factorials, read by antidiagonals.
9
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, 945, 280, 45, 6, 1, 1, 1, 5040, 10395, 3640, 585, 66, 7, 1, 1, 1, 40320, 135135, 58240, 9945, 1056, 91, 8, 1, 1, 1, 362880, 2027025, 1106560, 208845, 22176, 1729, 120, 9, 1, 1
OFFSET
0,8
COMMENTS
A variant of A142589.
FORMULA
A(n, k) = (-n)^k*FallingFactorial(-1/n, k) for n >= 1. - Peter Luschny, Dec 21 2021
EXAMPLE
1 1 1 1 1 1 1... A000012
1 1 2 6 24 120 720... A000142
1 1 3 15 105 945 10395... A001147
1 1 4 28 280 3640 58240... A007559
1 1 5 45 585 9945 208845... A007696
1 1 6 66 1056 22176 576576... A008548
1 1 7 91 1729 43225 1339975... A008542
1 1 8 120 2640 76560 2756160... A045754
1 1 9 153 3825 126225 5175225... A045755
1 1 10 190 5320 196840 9054640... A045756
1 1 11 231 7161 293601 14977651... A144773
MAPLE
seq(seq( mul(j*k+1, j=0..n-k-1), k=0..n), n=0..12); # G. C. Greubel, Mar 04 2020
MATHEMATICA
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 *)
PROG
(PARI) T(n, k) = prod(j=0, n-1, j*k+1);
for(n=0, 12, for(k=0, n, print1(T(n-k, k), ", "))) \\ G. C. Greubel, Mar 04 2020
(Magma)
function T(n, k)
if k eq 0 or n eq 0 then return 1;
else return (&*[j*k+1: j in [0..n-1]]);
end if; return T; end function;
[T(n-k, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 04 2020
(SageMath) [[ 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
(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
CROSSREFS
Cf. Diagonals : A092985, A076111, A158887.
Cf. A000142 ("1-fold"), A001147 (2-fold), A007559 (3), A007696 (4), A008548 (5), A008542 (6), A045754 (7), A045755 (8), A045756 (9), A144773 (10)
Sequence in context: A335432 A229557 A332700 * A213275 A069777 A382858
KEYWORD
nonn,tabl
AUTHOR
Philippe Deléham, Jun 01 2015
STATUS
approved