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%I #24 Mar 09 2021 20:14:47
%S 1,1,1,1,2,3,1,3,7,13,1,4,13,34,73,1,5,21,73,209,501,1,6,31,136,501,
%T 1546,4051,1,7,43,229,1045,4051,13327,37633,1,8,57,358,1961,9276,
%U 37633,130922,394353,1,9,73,529,3393,19081,93289,394353,1441729,4596553
%N Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(x/(1-x))/(1-x)^k.
%H Seiichi Manyama, <a href="/A293985/b293985.txt">Antidiagonals n = 0..139, flattened</a>
%F A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} (j+k)*A(n-j,k)/(n-j)! for n > 0.
%F A(0,k) = 1, A(1,k) = k+1 and A(n,k) = (2*n-1+k)*A(n-1,k) - (n-1)*(n-2+k)*A(n-2,k) for n > 1.
%e Square array begins:
%e 1, 1, 1, 1, 1, ... A000012;
%e 1, 2, 3, 4, 5, ... A000027;
%e 3, 7, 13, 21, 31, ... A002061;
%e 13, 34, 73, 136, 229, ... A135859;
%e 73, 209, 501, 1045, 1961, ...
%e 501, 1546, 4051, 9276, 19081, ...
%e Antidiagonal rows begin as:
%e 1;
%e 1, 1;
%e 1, 2, 3;
%e 1, 3, 7, 13;
%e 1, 4, 13, 34, 73;
%e 1, 5, 21, 73, 209, 501; - _G. C. Greubel_, Mar 09 2021
%t t[n_, k_]:= t[n, k]= If[n==0, 1, (n-1)!*Sum[(j+k)*t[n-j,k]/(n-j)!, {j,n}]];
%t T[n_,k_]:= t[k,n-k]; Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 09 2021 *)
%o (Sage)
%o @CachedFunction
%o def t(n,k): return 1 if n==0 else factorial(n-1)*sum( (j+k)*t(n-j,k)/factorial(n-j) for j in (1..n) )
%o def T(n,k): return t(k,n-k)
%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 09 2021
%o (Magma)
%o function t(n,k)
%o if n eq 0 then return 1;
%o else return Factorial(n-1)*(&+[(j+k)*t(n-j,k)/Factorial(n-j): j in [1..n]]);
%o end if; return t;
%o end function;
%o [t(k,n-k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 09 2021
%Y Columns k=0..6 give: A000262, A002720, A000262(n+1), A052852(n+1), A062147, A062266, A062192.
%Y Main diagonal gives A152059.
%Y Similar table: A086885, A088699, A176120.
%K nonn,tabl
%O 0,5
%A _Seiichi Manyama_, Oct 21 2017
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