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 A320031 #30 Jun 27 2025 18:16:57
%S A320031 1,1,1,1,2,1,1,3,5,1,1,4,13,16,1,1,5,25,79,65,1,1,6,41,226,633,326,1,
%T A320031 1,7,61,493,2713,6331,1957,1,1,8,85,916,7889,40696,75973,13700,1,1,9,
%U A320031 113,1531,18321,157781,732529,1063623,109601,1,1,10,145,2374,36745,458026,3786745,15383110,17017969,986410,1
%N A320031 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of the e.g.f. exp(x)/(1 - k*x).
%H A320031 Alois P. Heinz, <a href="/A320031/b320031.txt">Antidiagonals n = 0..140, flattened</a>
%F A320031 E.g.f. of column k: exp(x)/(1 - k*x).
%F A320031 A(n,k) = Sum_{j=0..n} binomial(n,j)*j!*k^j.
%F A320031 A(n,k) = hypergeom_2F0([1, -n], [], -k).
%F A320031 A(n,k) = 1 + [n > 0] * k * n * A(n-1,k). - _Alois P. Heinz_, May 09 2020
%F A320031 A(n,k) = floor(n!*k^n*exp(1/k)), k > 0, n + k > 1. - _Peter McNair_, Dec 20 2021
%F A320031 From _Werner Schulte_, Apr 14 2024: (Start)
%F A320031 The LU decomposition of this array is given by the upper triangular matrix U which is the transpose of A007318 and the lower triangular matrix L = A371898, i.e., A(n, k) = Sum_{i=0..k} binomial(k, i) * A371898(n, i).
%F A320031 Conjecture: E.g.f. of row n is exp(x) * (Sum_{k=0..n} A371898(n, k) * x^k / k!). (End)
%e A320031 E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (2*k^2 + 2*k + 1)*x^2/2! + (6*k^3 + 6*k^2 + 3*k + 1)*x^3/3! + (24*k^4 + 24*k^3 + 12*k^2 + 4*k + 1)*x^4/4! + ...
%e A320031 Square array begins:
%e A320031 1, 1, 1, 1, 1, 1, ...
%e A320031 1, 2, 3, 4, 5, 6, ...
%e A320031 1, 5, 13, 25, 41, 61, ...
%e A320031 1, 16, 79, 226, 493, 916, ...
%e A320031 1, 65, 633, 2713, 7889, 18321, ...
%e A320031 1, 326, 6331, 40696, 157781, 458026, ...
%p A320031 A := (n, k) -> simplify(hypergeom([1, -n], [], -k)):
%p A320031 for n from 0 to 5 do seq(A(n, k), k=0..8) od; # _Peter Luschny_, Oct 03 2018
%p A320031 # second Maple program:
%p A320031 A:= proc(n, k) option remember;
%p A320031 1 + `if`(n>0, k*n*A(n-1, k), 0)
%p A320031 end:
%p A320031 seq(seq(A(n, d-n), n=0..d), d=0..12); # _Alois P. Heinz_, May 09 2020
%t A320031 Table[Function[k, n! SeriesCoefficient[Exp[x]/(1 - k x), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%t A320031 Table[Function[k, HypergeometricPFQ[{1, -n}, {}, -k]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%Y A320031 Columns k=0..6 give A000012, A000522, A010844, A010845, A056545, A056546, A056547.
%Y A320031 Main diagonal gives A277452.
%Y A320031 Cf. A007318, A320032, A371898.
%K A320031 nonn,tabl
%O A320031 0,5
%A A320031 _Ilya Gutkovskiy_, Oct 03 2018