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

Author

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

Keywords

• nonn
• tabl

Comments

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

Examples

			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
		

Links

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

Crossrefs

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

Programs

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

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

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

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

Extensions

Edited by M. F. Hasler, Oct 28 2014

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