cp's OEIS Frontend

A007776 Number of connected posets with n elements of height 1.

%I A007776 #40 Nov 29 2023 11:42:04
%S A007776 1,2,4,10,27,88,328,1460,7799,51196,422521,4483460,62330116,
%T A007776 1150504224,28434624153,945480850638,42417674401330,2572198227615998,
%U A007776 211135833162079184,23487811567341121158,3545543330739039981738,727053904070651775719646
%N A007776 Number of connected posets with n elements of height 1.
%C A007776 Inverse Euler transform of A048194 and A049312. - Detlef Pauly (dettodet(AT)yahoo.de) and _Vladeta Jovovic_, Jul 25 2003
%C A007776 Essentially the same as A318870. - _Georg Fischer_, Oct 02 2018
%C A007776 Number of connected digraphs on n unlabeled nodes where every node has indegree 0 or outdegree 0 and there are no isolated nodes.  - _Andrew Howroyd_, Oct 03 2018
%H A007776 Andrew Howroyd, <a href="/A007776/b007776.txt">Table of n, a(n) for n = 2..50</a> (terms 2..40 from Alois P. Heinz)
%H A007776 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A007776 J. Textor, A. Idelberger, and M. Liskiewicz, <a href="https://arxiv.org/abs/1508.00280">Learning from Pairwise Marginal Independencies</a>, arXiv:1508.00280 [cs.AI], 2015.
%H A007776 <a href="/index/Pos#posets">Index entries for sequences related to posets</a>
%F A007776 Inverse Euler transform of A055192. - _Andrew Howroyd_, Oct 03 2018
%t A007776 mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
%t A007776 EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
%t A007776 b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Flatten @ Table[Map[ Function[{p}, p + j*x^i], b[n - i*j, i - 1]], {j, 0, n/i}]]];
%t A007776 g[n_, k_] := g[n, k] = Sum[Sum[2^Sum[Sum[GCD[i, j]*Coefficient[s, x, i]* Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/ Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n + k, n + k]}], {s, b[n, n]}];
%t A007776 A[n_, k_] := g[Min[n, k], Abs[n - k]];
%t A007776 b[d_] := Sum[A[n, d - n], {n, 0, d}];
%t A007776 EULERi[Array[b, 30]] // Rest (* _Jean-François Alcover_, Sep 16 2019, after _Alois P. Heinz_ in A049312 *)
%Y A007776 Cf. A005142, A002031 (labeled case), A048194, A049312, A055192, A318870, column 1 of A342500.
%K A007776 nonn
%O A007776 2,2
%A A007776 Georg Wambach (gw(AT)informatik.Uni-Koeln.de)
%E A007776 More terms from _Vladeta Jovovic_, Jul 25 2003
%E A007776 Offset corrected by _Andrew Howroyd_, Oct 03 2018