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 A338435 #44 Feb 16 2025 08:34:00
%S A338435 1,1,1,1,2,2,1,3,14,6,1,4,34,168,24,1,5,62,654,2840,120,1,6,98,1626,
%T A338435 17688,61870,720,1,7,142,3246,59928,616120,1649232,5040,1,8,194,5676,
%U A338435 151064,2844120,26252496,51988748,40320,1,9,254,9078,318744,9052120,165100752,1322624016,1891712384,362880
%N A338435 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = n!*LaguerreL(n, -k*n).
%H A338435 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%H A338435 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A338435 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A338435 T(n,k) = Sum_{j=0..n} (k*n)^j * (n-j)! * binomial(n,j)^2.
%F A338435 T(n,k) = n! * [x^n] exp(k*n*x/(1-x))/(1-x).
%F A338435 T(n,k) = A289192(n,k*n).
%e A338435 Square array begins:
%e A338435 1, 1, 1, 1, 1, ...
%e A338435 1, 2, 3, 4, 5, ...
%e A338435 2, 14, 34, 62, 98, ...
%e A338435 6, 168, 654, 1626, 3246, ...
%e A338435 24, 2840, 17688, 59928, 151064, ...
%t A338435 T[n_, k_] := n! * LaguerreL[n, -k*n]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Feb 05 2021 *)
%o A338435 (PARI) T(n, k) = sum(j=0, n, (k*n)^j*(n-j)!*binomial(n, j)^2);
%o A338435 (PARI) T(n, k) = n!*pollaguerre(n, 0, -k*n); \\ _Michel Marcus_, Feb 05 2021
%Y A338435 Columns k=0..8 gives A000142, A277373, A277391, A277392, A277418, A277419, A277420, A277421, A277422.
%Y A338435 Main diagonal gives A340863.
%Y A338435 Cf. A021009, A289192 (n!*LaguerreL(n, -k)), A341014.
%K A338435 nonn,tabl
%O A338435 0,5
%A A338435 _Seiichi Manyama_, Feb 05 2021