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 A134558 #28 Jan 03 2021 15:56:09
%S A134558 1,1,1,2,2,1,6,5,3,1,24,16,10,4,1,120,65,38,17,5,1,720,326,168,78,26,
%T A134558 6,1,5040,1957,872,393,142,37,7,1,40320,13700,5296,2208,824,236,50,8,
%U A134558 1,362880,109601,37200,13977,5144,1569,366,65,9,1,3628800,986410,297856
%N A134558 Array read by antidiagonals, a(n,k) = gamma(n+1,k)*e^k, where gamma(n,k) is the upper incomplete gamma function and e is the exponential constant 2.71828...
%H A134558 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IncompleteGammaFunction.html">Incomplete Gamma Function</a>.
%H A134558 Wikipedia, <a href="https://en.wikipedia.org/wiki/Incomplete_gamma_function#Upper_incomplete_Gamma_function">Incomplete gamma function</a>.
%F A134558 a(n,k) = gamma(n+1,k)*e^k = Sum_{m=0..n} m!*binomial(n,m)*k^(n-m).
%F A134558 a(n,k) = n*a(n-1,k) + k^n for n,k > 0.
%F A134558 E.g.f. (by columns) is e^(kx)/(1-x).
%F A134558 a(n,k) = the binomial transform by columns of a(n,k-1).
%F A134558 Conjecture: a(n,k) is the permanent of the n X n matrix with k+1 on the main diagonal and 1 elsewhere.
%e A134558 Square array begins:
%e A134558 1, 1, 1, 1, 1, 1, 1, ...
%e A134558 1, 2, 3, 4, 5, 6, 7, ...
%e A134558 2, 5, 10, 17, 26, 37, 50, ...
%e A134558 6, 16, 38, 78, 142, 236, 366, ...
%e A134558 24, 65, 168, 393, 824, 1569, 2760, ...
%e A134558 120, 326, 872, 2208, 5144, 10970, 21576, ...
%e A134558 720, 1957, 5296, 13977, 34960, 81445, 176112, ...
%t A134558 T[n_,k_] := Gamma[n+1, k]*E^k; Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] //Flatten (* _Amiram Eldar_, Jun 27 2020 *)
%Y A134558 Cf. a(n, 0) = A000142(n); a(n, 1) = A000522(n); a(n, 2) = A010842(n); a(n, 3) = A053486(n); a(n, 4) = A053487(n); a(n, 5) = A080954(n); a(n, 6) = A108869(n); a(1, k) = A000027(k+1); a(2, k) = A002522(k+1); a(n, n) = A063170(n); a(n, n+1) = A001865(n+1); a(n, n+2) = A001863(n+2).
%Y A134558 Another version: A089258.
%Y A134558 A transposed version: A080955.
%Y A134558 Cf. A001113.
%K A134558 nonn,tabl
%O A134558 0,4
%A A134558 _Ross La Haye_, Jan 22 2008
%E A134558 More terms from _Amiram Eldar_, Jun 27 2020