Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #28 Apr 14 2024 11:49:34
%S 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 1,7,61,493,2713,6331,1957,1,1,8,85,916,7889,40696,75973,13700,1,1,9,
%U 113,1531,18321,157781,732529,1063623,109601,1,1,10,145,2374,36745,458026,3786745,15383110,17017969,986410,1
%N 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 Alois P. Heinz, <a href="/A320031/b320031.txt">Antidiagonals n = 0..140, flattened</a>
%F E.g.f. of column k: exp(x)/(1 - k*x).
%F A(n,k) = Sum_{j=0..n} binomial(n,j)*j!*k^j.
%F A(n,k) = hypergeom_2F0([1, -n], [], -k).
%F A(n,k) = 1 + [n > 0] * k * n * A(n-1,k). - _Alois P. Heinz_, May 09 2020
%F A(n,k) = floor(n!*k^n*exp(1/k)), k > 0, n + k > 1. - _Peter McNair_, Dec 20 2021
%F From _Werner Schulte_, April 14 2024: (Start)
%F 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 Conjecture: E.g.f. of row n is exp(x) * (Sum_{k=0..n} A371898(n, k) * x^k / k!). (End)
%e 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 Square array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 3, 4, 5, 6, ...
%e 1, 5, 13, 25, 41, 61, ...
%e 1, 16, 79, 226, 493, 916, ...
%e 1, 65, 633, 2713, 7889, 18321, ...
%e 1, 326, 6331, 40696, 157781, 458026, ...
%p A := (n, k) -> simplify(hypergeom([1, -n], [], -k)):
%p for n from 0 to 5 do seq(A(n, k), k=0..8) od; # _Peter Luschny_, Oct 03 2018
%p # second Maple program:
%p A:= proc(n, k) option remember;
%p 1 + `if`(n>0, k*n*A(n-1, k), 0)
%p end:
%p seq(seq(A(n, d-n), n=0..d), d=0..12); # _Alois P. Heinz_, May 09 2020
%t Table[Function[k, n! SeriesCoefficient[Exp[x]/(1 - k x), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%t Table[Function[k, HypergeometricPFQ[{1, -n}, {}, -k]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
%Y Columns k=0..6 give A000012, A000522, A010844, A010845, A056545, A056546, A056547.
%Y Main diagonal gives A277452.
%Y Cf. A007318, A320032, A371898.
%K nonn,tabl
%O 0,5
%A _Ilya Gutkovskiy_, Oct 03 2018