A142589 Square array T(n,m) = Product_{i=0..m} (1+n*i) read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 24, 15, 4, 1, 1, 120, 105, 28, 5, 1, 1, 720, 945, 280, 45, 6, 1, 1, 5040, 10395, 3640, 585, 66, 7, 1, 1, 40320, 135135, 58240, 9945, 1056, 91, 8, 1, 1, 362880, 2027025, 1106560, 208845, 22176, 1729, 120, 9, 1, 1, 3628800, 34459425, 24344320, 5221125, 576576, 43225, 2640, 153, 10, 1

Offset

0,5

Comments

Antidiagonal sums are {1, 2, 4, 11, 45, 260, 1998, 19735, 244797, 3729346, 68276276, ...}.

Links

G. C. Greubel, Antidiagonal rows n = 0..100, flattened

Example

The transpose of the array is:
    1,    1,     1,     1,      1,      1,      1,      1,     1,
    1,    2,     3,     4,      5,      6,      7,      8,     9,
    1,    6,    15,    28,     45,     66,     91,     120,   153, ... A000384
    1,   24,   105,   280,    585,   1056,   1729,    2640,  3825, ... A011199
    1,  120,   945,  3640,   9945,  22176,  43225,   76560, 126225,... A011245
    1,  720, 10395, 58240, 208845, 576576, 1339975, 2756160,...
        /      |       \       \
   A000142  A001147  A007559  A007696

Maple

T:= (n, k)-> `if`(n=0, 1, mul(j*k+1, j=0..n)):
seq(seq(T(n-k, k), k=0..n), n=0..12); # G. C. Greubel, Mar 05 2020

Mathematica

T[n_, k_]= If[n==0, 1, Product[1 + k*i, {i, 0, n}]]; Table[T[n-k, k], {n, 0, 10}, {k, 0, n}]//Flatten

Prog

(PARI) T(n, k) = if(n==0, 1, prod(j=0, n, j*k+1) );
for(n=0, 12, for(k=0, n, print1(T(n-k, k), ", "))) \\ G. C. Greubel, Mar 05 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]]);
  end if; return T; end function;
[T(n-k, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 05 2020
(Sage)
def T(n, k):
    if (k==0 and n==0): return 1
    else: return product(j*k+1 for j in (0..n))
[[T(n-k, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 05 2020

Crossrefs

Cf. A000142, A006882(2n-1) = A001147, A007661(3n-2) = A007559, A007662(4n-3) = A007696, A153274.

Sequence in context: A181644 A144351 A213936 * A284308 A369435 A172400

Adjacent sequences: A142586 A142587 A142588 * A142590 A142591 A142592

Keyword

nonn,tabl

Author

Roger L. Bagula and Gary W. Adamson, Sep 22 2008

Extensions

Edited by M. F. Hasler, Oct 28 2014

More terms added by G. C. Greubel, Mar 05 2020

Status

approved