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