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

1, 1, 7, 75, 1105, 20821, 478439, 12977815, 405909913, 14382249193, 569377926495, 24908595049347, 1193272108866953, 62128556769033261, 3493232664307133871, 210943871609662171055, 13615857409567572389361, 935523911378273899335537

Offset: 0

Author

Goran Kilibarda and Vladeta Jovovic, Aug 25 2006

Keywords

• nonn

Comments

Also number of labeled multigraphs without isolated vertices and with n edges.

Links

• Nathaniel Johnston, Table of n, a(n) for n = 0..125
• A. N. Bhavale, B. N. Waphare, Basic retracts and counting of lattices, Australasian J. of Combinatorics (2020) Vol. 78, No. 1, 73-99.

Crossrefs

Row n=2 of A330942.

Cf. A055203, A121251, A104209.

Programs

Maple

seq(sum(binomial(k*(k-1)/2+n-1,n)/2^(k+1),k=0..infinity),n=0..20);
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

Mathematica

Table[Sum[Binomial[k*(k-1)/2+n-1,n]/2^(k+1),{k,0,Infinity}],{n,0,20}] (* Vaclav Kotesovec, Mar 15 2014 *)

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

Formula

a(n) = (1/n!)* Sum_{k=0..n} |Stirling1(n,k)|*A055203(k).

a(n) = Sum_{k>=0} binomial(k*(k-1)/2+n-1,n)/2^(k+1).

a(n) ~ n^n * 2^(n-1 + log(2)/4) / (exp(n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Mar 15 2014

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