A218112 Number of transitive reflexive early confluent binary relations R on n+2 labeled elements with max_{x}(|{y : xRy}|) = n.

0, 1, 61, 1105, 16025, 239379, 3794378, 64432638, 1173919350, 22913136730, 477859512889, 10616510910603, 250501631648359, 6259150585043685, 165157651772590340, 4590337237739801932, 134066099253229461636, 4105495811166963962292, 131552972087266209052875

Offset: 0

Author

Alois P. Heinz, Oct 20 2012

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200 (terms n=23..100 from Vincenzo Librandi)

Programs

Maple

t:= proc(k) option remember; `if`(k<0, 0, unapply(exp(add(x^m/m! *t(k-m)(x), m=1..k)), x)) end: tt:= proc(k) option remember; unapply((t(k)-t(k-1))(x), x) end: T:= proc(n, k) option remember; coeff(series(tt(k)(x), x, n+1), x, n) *n! end:
a:= n-> T(n+2,n): seq(a(n), n=0..20);

Mathematica

t[k_] := t[k] = If[k < 0, 0&, Function[x, Evaluate @ Normal[Series[Exp[Sum[x^m/m!*t[k-m][x], {m, 1, k}]], {x, 0, k+3}]]]]; tt[k_] := tt[k] = Function[x, (t[k][x]-t[k-1][x]) // Evaluate]; T[n_, k_] := T[n, k] = Coefficient[Series[tt[k][x], {x, 0, n+1}], x, n]*n!; a[n_] := a[n] = T[n+2, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 17 2014, after Maple *)

Formula

a(n) = A135313(n+2,n).

a(n) ~ n! * n^4 / (16 * log(2)^(n+3)). - Vaclav Kotesovec, Nov 20 2021