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

0,3

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.

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

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 *)

Prog

(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

Crossrefs

Row n=2 of A330942.

Cf. A055203, A121251, A104209.

Sequence in context: A243692 A258293 A127190 * A220215 A066302 A106162

Adjacent sequences: A121313 A121314 A121315 * A121317 A121318 A121319

Keyword

nonn

Author

Goran Kilibarda and Vladeta Jovovic, Aug 25 2006

Status

approved