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A293985 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.

%I A293985 #37 Jan 27 2025 11:39:10
%S A293985 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 A293985 1546,4051,1,7,43,229,1045,4051,13327,37633,1,8,57,358,1961,9276,
%U A293985 37633,130922,394353,1,9,73,529,3393,19081,93289,394353,1441729,4596553
%N A293985 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 A293985 Seiichi Manyama, <a href="/A293985/b293985.txt">Antidiagonals n = 0..139, flattened</a>
%H A293985 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A293985 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A293985 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 A293985 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.
%F A293985 From _Seiichi Manyama_, Jan 25 2025: (Start)
%F A293985 A(n,k) = n! * Sum_{j=0..n} binomial(n+k-1,j)/(n-j)!.
%F A293985 A(n,k) = n! * LaguerreL(n, k-1, -1). (End)
%e A293985 Square array begins:
%e A293985     1,    1,    1,    1,     1, ... A000012;
%e A293985     1,    2,    3,    4,     5, ... A000027;
%e A293985     3,    7,   13,   21,    31, ... A002061;
%e A293985    13,   34,   73,  136,   229, ... A135859;
%e A293985    73,  209,  501, 1045,  1961, ...
%e A293985   501, 1546, 4051, 9276, 19081, ...
%e A293985 Antidiagonal rows begin as:
%e A293985   1;
%e A293985   1, 1;
%e A293985   1, 2,  3;
%e A293985   1, 3,  7, 13;
%e A293985   1, 4, 13, 34,  73;
%e A293985   1, 5, 21, 73, 209, 501; - _G. C. Greubel_, Mar 09 2021
%t A293985 t[n_, k_]:= t[n, k]= If[n==0, 1, (n-1)!*Sum[(j+k)*t[n-j,k]/(n-j)!, {j,n}]];
%t A293985 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 A293985 (Sage)
%o A293985 @CachedFunction
%o A293985 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 A293985 def T(n,k): return t(k,n-k)
%o A293985 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 09 2021
%o A293985 (Magma)
%o A293985 function t(n,k)
%o A293985   if n eq 0 then return 1;
%o A293985   else return Factorial(n-1)*(&+[(j+k)*t(n-j,k)/Factorial(n-j): j in [1..n]]);
%o A293985   end if; return t;
%o A293985 end function;
%o A293985 [t(k,n-k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 09 2021
%Y A293985 Columns k=0..6 give: A000262, A002720, A000262(n+1), A052852(n+1), A062147, A062266, A062192.
%Y A293985 Main diagonal gives A152059.
%Y A293985 Similar table: A086885, A088699, A176120.
%K A293985 nonn,tabl
%O A293985 0,5
%A A293985 _Seiichi Manyama_, Oct 21 2017