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 A342186 #45 Mar 05 2025 11:38:10
%S A342186 1,-1,1,3,-4,1,-21,31,-11,1,315,-486,196,-26,1,-9765,15381,-6562,1002,
%T A342186 -57,1,615195,-978768,428787,-69688,4593,-120,1,-78129765,124918731,
%U A342186 -55434717,9279163,-652999,19833,-247,1
%N A342186 Triangle read by rows, matrix inverse of A139382.
%C A342186 This triangle appears to be the q-analog of A008275 (Stirling numbers of the first kind) for q=2. However, A333142 has a similar definition.
%C A342186 Row sums of unsigned triangle are A006125 with offset 1.
%C A342186 |T(n,k)| is the number of descent free digraphs on [n] containing exactly k source nodes. A descent in a digraph is a pair of vertices s->t such that s>t. A descent free digraph is necessarily acyclic. A source in an acyclic digraph is a node with indegree 0. - _Geoffrey Critzer_, Mar 05 2025
%H A342186 John Keith, <a href="/A342186/b342186.txt">Rows n = 1..20 of triangle, flattened</a>
%H A342186 Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, <a href="https://arxiv.org/abs/1909.01550">Counting acyclic and strong digraphs by descents</a>, arXiv:1909.01550 [math.CO], 20 Mar 2020.
%F A342186 T(n,k) = T(n-1,k-1) - (2^(n-1)-1) * T(n-1,k), n, k >= 1, T(1, 1) = 1, T(n, 0) = 0.
%F A342186 For unsigned triangle, T(n, 1) = A005329(n-1); T(n, 2) = A203011(n-1); T(n, n-1) = A000295(n+1); T(n, n-2) = A203242(n-1).
%F A342186 T(n,k) = Sum_{j=k..n} (-1)^(n-j)*2^binomial(n-j,2)*qBinomial(n,j,2)*binomial(j,k), where qBinomial(n,k,2) is A022166(n,k). - _Fabian Pereyra_, Feb 08 2024
%e A342186 The triangle begins:
%e A342186 1;
%e A342186 -1, 1;
%e A342186 3, -4, 1;
%e A342186 -21, 31, -11, 1;
%e A342186 315, -486, 196, -26, 1;
%e A342186 -9765, 15381, -6562, 1002, -57, 1;
%e A342186 615195, -978768, 428787, -69688, 4593, -120, 1;
%e A342186 -78129765, 124918731, -55434717, 9279163, -652999, 19833, -247, 1;
%e A342186 ...
%p A342186 A342186 := proc(n, k) if n = 1 and k = 1 then 1 elif k > n or k < 1 then 0 else
%p A342186 A342186(n-1, k-1) - (2^(n-1) - 1) * A342186(n-1, k) fi end:
%p A342186 for n from 1 to 8 do seq(A342186(n, k), k = 1..n) od; # _Peter Luschny_, Jun 28 2022
%t A342186 T[1, 1] := 1; T[n_, k_] := T[n, k] = If[k > n || k < 1, 0, T[n - 1, k - 1] - (2^(n - 1) - 1)*T[n - 1, k]]; Table[T[n, k], {n, 1, 8}, {k, 1, n}] (* after _G. C. Greubel_'s program for A139382 *)
%o A342186 (PARI) mat(nn) = my(m = matrix(nn, nn)); for (n=1, nn, for(k=1, nn, m[n,k] = if (n==1, if (k==1, 1, 0), if (k==1, 1, (2^k-1)*m[n-1, k] + m[n-1, k-1])););); m; \\ A139382
%o A342186 tabl(nn) = 1/mat(nn); \\ _Michel Marcus_, Mar 18 2021
%Y A342186 Cf. A008275, A139382, A333142, A333143, A006125 (row sums).
%Y A342186 Columns of unsigned triangle: A005329, A203011, A000295, A203242.
%K A342186 sign,tabl
%O A342186 1,4
%A A342186 _John Keith_, Mar 04 2021