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 A355440 #19 Aug 24 2023 07:49:02
%S A355440 1,2,10,98,2050,84482,7221250,1218502658,421846581250,288641130823682,
%T A355440 403002184457781250,1112950376623239069698,6251793960501383945781250,
%U A355440 69503063309910921346390425602,1568447691296998939150390025781250
%N A355440 Expansion of e.g.f. Sum_{k>=0} exp(4^k * x) * x^k/k!.
%C A355440 a(n) is the number of labeled digraphs on [n] with self loops allowed (A002416) such that reflexive nodes are only adjacent to irreflexive nodes and vice versa. A reflexive node is a node with a self loop. An irreflexive node is a node without a self loop. - _Geoffrey Critzer_, Aug 21 2023
%F A355440 G.f.: Sum_{k>=0} x^k/(1 - 4^k * x)^(k+1).
%F A355440 a(n) = Sum_{k=0..n} 4^(k*(n-k)) * binomial(n,k).
%F A355440 E(x)^2 = Sum_{k>=0} a(n)*x^n/B(n) where B(n) = n!*4^binomial(n,2) and E(x) = Sum_{n>=0} x^n/B(n). - _Geoffrey Critzer_, Aug 21 2023
%t A355440 nn = 6; B[n_] := n! 4^Binomial[n, 2]; e[x_] := Sum[x^n/B[n], {n, 0, nn}];
%t A355440 Table[B[n], {n, 0, nn}] CoefficientList[Series[e[x]^2, {x, 0, nn}], x]
%o A355440 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, exp(4^k*x)*x^k/k!)))
%o A355440 (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-4^k*x)^(k+1)))
%o A355440 (PARI) a(n) = sum(k=0, n, 4^(k*(n-k))*binomial(n, k));
%Y A355440 Column k=4 of A355395.
%Y A355440 Cf. A193199.
%K A355440 nonn
%O A355440 0,2
%A A355440 _Seiichi Manyama_, Jul 02 2022