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A121316 Unlabeled version of A055203: number of different relations between n intervals (of nonzero length) on a line, up to permutation of intervals.

%I A121316 #18 Oct 15 2020 16:34:33
%S A121316 1,1,7,75,1105,20821,478439,12977815,405909913,14382249193,
%T A121316 569377926495,24908595049347,1193272108866953,62128556769033261,
%U A121316 3493232664307133871,210943871609662171055,13615857409567572389361,935523911378273899335537
%N A121316 Unlabeled version of A055203: number of different relations between n intervals (of nonzero length) on a line, up to permutation of intervals.
%C A121316 Also number of labeled multigraphs without isolated vertices and with n edges.
%H A121316 Nathaniel Johnston, <a href="/A121316/b121316.txt">Table of n, a(n) for n = 0..125</a>
%H A121316 A. N. Bhavale, B. N. Waphare, <a href="https://ajc.maths.uq.edu.au/pdf/78/ajc_v78_p073.pdf">Basic retracts and counting of lattices</a>, Australasian J. of Combinatorics (2020) Vol. 78, No. 1, 73-99.
%F A121316 a(n) = (1/n!)* Sum_{k=0..n} |Stirling1(n,k)|*A055203(k).
%F A121316 a(n) = Sum_{k>=0} binomial(k*(k-1)/2+n-1,n)/2^(k+1).
%F A121316 a(n) ~ n^n * 2^(n-1 + log(2)/4) / (exp(n) * (log(2))^(2*n+1)). - _Vaclav Kotesovec_, Mar 15 2014
%F A121316 a(n) = Sum_{j=0..2*n} binomial(binomial(j,2)+n-1, n) * (Sum_{i=j..2*n} (-1)^(i-j)*binomial(i,j)). - _Andrew Howroyd_, Feb 09 2020
%p A121316 seq(sum(binomial(k*(k-1)/2+n-1,n)/2^(k+1),k=0..infinity),n=0..20);
%p A121316 with(combinat): A121316:=proc(n) return (1/n!)*add(abs(stirling1(n,k))*A055203(k),k=0..n): end: seq(A121316(n),n=0..20); # _Nathaniel Johnston_, Apr 28 2011
%t A121316 Table[Sum[Binomial[k*(k-1)/2+n-1,n]/2^(k+1),{k,0,Infinity}],{n,0,20}] (* _Vaclav Kotesovec_, Mar 15 2014 *)
%o A121316 (PARI) a(n) = {sum(j=0, 2*n, binomial(binomial(j,2)+n-1, n) * sum(i=j, 2*n, (-1)^(i-j)*binomial(i,j)))} \\ _Andrew Howroyd_, Feb 09 2020
%Y A121316 Row n=2 of A330942.
%Y A121316 Cf. A055203, A121251, A104209.
%K A121316 nonn
%O A121316 0,3
%A A121316 Goran Kilibarda and _Vladeta Jovovic_, Aug 25 2006