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

%I A142589 #20 Sep 08 2022 08:45:36
%S A142589 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,
%T A142589 1,1,5040,10395,3640,585,66,7,1,1,40320,135135,58240,9945,1056,91,8,1,
%U A142589 1,362880,2027025,1106560,208845,22176,1729,120,9,1,1,3628800,34459425,24344320,5221125,576576,43225,2640,153,10,1
%N A142589 Square array T(n,m) = Product_{i=0..m} (1+n*i) read by antidiagonals.
%C A142589 Antidiagonal sums are {1, 2, 4, 11, 45, 260, 1998, 19735, 244797, 3729346, 68276276, ...}.
%H A142589 G. C. Greubel, <a href="/A142589/b142589.txt">Antidiagonal rows n = 0..100, flattened</a>
%e A142589 The transpose of the array is:
%e A142589     1,    1,     1,     1,      1,      1,      1,      1,     1,
%e A142589     1,    2,     3,     4,      5,      6,      7,      8,     9,
%e A142589     1,    6,    15,    28,     45,     66,     91,     120,   153, ... A000384
%e A142589     1,   24,   105,   280,    585,   1056,   1729,    2640,  3825, ... A011199
%e A142589     1,  120,   945,  3640,   9945,  22176,  43225,   76560, 126225,... A011245
%e A142589     1,  720, 10395, 58240, 208845, 576576, 1339975, 2756160,...
%e A142589         /      |       \       \
%e A142589    A000142  A001147  A007559  A007696
%p A142589 T:= (n, k)-> `if`(n=0, 1, mul(j*k+1, j=0..n)):
%p A142589 seq(seq(T(n-k, k), k=0..n), n=0..12); # _G. C. Greubel_, Mar 05 2020
%t A142589 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
%o A142589 (PARI) T(n, k) = if(n==0, 1, prod(j=0, n, j*k+1) );
%o A142589 for(n=0, 12, for(k=0, n, print1(T(n-k, k), ", "))) \\ _G. C. Greubel_, Mar 05 2020
%o A142589 (Magma)
%o A142589 function T(n,k)
%o A142589   if k eq 0 or n eq 0 then return 1;
%o A142589   else return (&*[j*k+1: j in [0..n]]);
%o A142589   end if; return T; end function;
%o A142589 [T(n-k,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 05 2020
%o A142589 (Sage)
%o A142589 def T(n, k):
%o A142589     if (k==0 and n==0): return 1
%o A142589     else: return product(j*k+1 for j in (0..n))
%o A142589 [[T(n-k, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Mar 05 2020
%Y A142589 Cf. A000142, A006882(2n-1) = A001147, A007661(3n-2) = A007559, A007662(4n-3) = A007696, A153274.
%K A142589 nonn,tabl
%O A142589 0,5
%A A142589 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 22 2008
%E A142589 Edited by _M. F. Hasler_, Oct 28 2014
%E A142589 More terms added by _G. C. Greubel_, Mar 05 2020