This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A343847 #24 May 09 2021 08:04:13
%S A343847 1,1,1,2,2,1,6,7,3,1,24,34,14,4,1,120,209,86,23,5,1,720,1546,648,168,
%T A343847 34,6,1,5040,13327,5752,1473,286,47,7,1,40320,130922,58576,14988,2840,
%U A343847 446,62,8,1,362880,1441729,671568,173007,32344,4929,654,79,9,1
%N A343847 T(n, k) = (n - k)! * [x^(n-k)] exp(k*x/(1 - x))/(1 - x). Triangle read by rows, T(n, k) for 0 <= k <= n.
%F A343847 T(n, k) = (-1)^(n - k)*U(k - n, 1, -k), where U is the Kummer U function.
%F A343847 T(n, k) = (n - k)! * L(n - k, -k), where L is the Laguerre polynomial function.
%F A343847 T(n, k) = (n - k)! * Sum_{j = 0..n - k} binomial(n - k, j) k^j / j!.
%F A343847 T(n, k) = (2*n-k-1)*T(n-1, k) - (n-k-1)^2*T(n-2, k) for n - k >= 2.
%e A343847 Triangle starts:
%e A343847 0: 1;
%e A343847 1: 1, 1;
%e A343847 2: 2, 2, 1;
%e A343847 3: 6, 7, 3, 1;
%e A343847 4: 24, 34, 14, 4, 1;
%e A343847 5: 120, 209, 86, 23, 5, 1;
%e A343847 6: 720, 1546, 648, 168, 34, 6, 1;
%e A343847 7: 5040, 13327, 5752, 1473, 286, 47, 7, 1;
%e A343847 8: 40320, 130922, 58576, 14988, 2840, 446, 62, 8, 1;
%e A343847 .
%e A343847 Array whose upward read antidiagonals are the rows of the triangle.
%e A343847 n\k 0 1 2 3 4 5 6
%e A343847 -----------------------------------------------------------------
%e A343847 0: 1, 1, 1, 1, 1, 1, 1, ...
%e A343847 1: 1, 2, 3, 4, 5, 6, 7, ...
%e A343847 2: 2, 7, 14, 23, 34, 47, 62, ...
%e A343847 3: 6, 34, 86, 168, 286, 446, 654, ...
%e A343847 4: 24, 209, 648, 1473, 2840, 4929, 7944, ...
%e A343847 5: 120, 1546, 5752, 14988, 32344, 61870, 108696, ...
%e A343847 6: 720, 13327, 58576, 173007, 414160, 866695, 1649232, ...
%e A343847 7: 5040, 130922, 671568, 2228544, 5876336, 13373190, 27422352, ...
%p A343847 T := proc(n, k) option remember;
%p A343847 if n = k then return 1 elif n = k+1 then return k+1 fi;
%p A343847 (2*n-k-1)*T(n-1, k) - (n-k-1)^2*T(n-2, k) end:
%p A343847 seq(print(seq(T(n ,k), k = 0..n)), n = 0..7);
%t A343847 T[n_, k_] := (-1)^(n - k) HypergeometricU[k - n, 1, -k];
%t A343847 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
%t A343847 (* Alternative: *)
%t A343847 TL[n_, k_] := (n - k)! LaguerreL[n - k, -k];
%t A343847 Table[TL[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
%o A343847 (PARI)
%o A343847 T(n, k) = (n - k)!*sum(j=0, n - k, binomial(n - k, j) * k^j / j!)
%o A343847 for(n=0, 9, for(k=0, n, print(T(n, k))))
%o A343847 (SageMath) # Columns of the array.
%o A343847 def column(k, len):
%o A343847 R.<x> = PowerSeriesRing(QQ, default_prec=len)
%o A343847 f = exp(k * x / (1 - x)) / (1 - x)
%o A343847 return f.egf_to_ogf().list()
%o A343847 for col in (0..6): print(column(col, 20))
%Y A343847 Row sums: A343848. T(2*n, n) = A277373(n). Variant: A289192.
%Y A343847 Columns: A000142, A002720, A087912, A277382, A289147, A289211, A289212, A289213, A289214, A289215, A289216.
%Y A343847 Cf. A021009 (Laguerre polynomials), A344048.
%K A343847 nonn,tabl
%O A343847 0,4
%A A343847 _Peter Luschny_, May 07 2021