A218112 - OEIS

%I #21 Nov 20 2021 10:01:49

%S 0,1,61,1105,16025,239379,3794378,64432638,1173919350,22913136730,

%T 477859512889,10616510910603,250501631648359,6259150585043685,

%U 165157651772590340,4590337237739801932,134066099253229461636,4105495811166963962292,131552972087266209052875

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

%H Alois P. Heinz, <a href="/A218112/b218112.txt">Table of n, a(n) for n = 0..200</a> (terms n=23..100 from Vincenzo Librandi)

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

%F a(n) ~ n! * n^4 / (16 * log(2)^(n+3)). - _Vaclav Kotesovec_, Nov 20 2021

%p 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:

%p a:= n-> T(n+2,n): seq(a(n), n=0..20);

%t 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 *)

%K nonn

%O 0,3

%A _Alois P. Heinz_, Oct 20 2012