A218111
Number of transitive reflexive early confluent binary relations R on n+1 labeled elements with max_{x}(|{y : xRy}|) = n.
Original entry on oeis.org
0, 1, 12, 106, 1035, 11301, 137774, 1863044, 27733869, 451238935, 7972318200, 152065270974, 3115418734415, 68245059703289, 1591993733475570, 39406010771574856, 1031649940977825633, 28483179899706237483, 827159099070697636124, 25205610503231757308450
Offset: 0
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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+1,n): seq(a(n), n=0..20);
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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+2}]]]]; 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+1, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 17 2014, after Maple *)
A218112
Number of transitive reflexive early confluent binary relations R on n+2 labeled elements with max_{x}(|{y : xRy}|) = n.
Original entry on oeis.org
0, 1, 61, 1105, 16025, 239379, 3794378, 64432638, 1173919350, 22913136730, 477859512889, 10616510910603, 250501631648359, 6259150585043685, 165157651772590340, 4590337237739801932, 134066099253229461636, 4105495811166963962292, 131552972087266209052875
Offset: 0
-
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);
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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 *)
A261238
Number of transitive reflexive early confluent binary relations R on 2n labeled elements where max_{x}(|{y:xRy}|)=n.
Original entry on oeis.org
1, 1, 61, 12075, 4798983, 3151808478, 3085918099231, 4210378306984993, 7631859877504516225, 17735784941946000072572, 51404873131596488549863350, 181773929944698613445522139632, 770224297920086034338727292711511, 3852558194920465350481058381000064850
Offset: 0
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t:= proc(k) option remember; `if`(k<0, 0,
exp(add(x^m/m!*t(k-m), m=1..k)))
end:
A:= proc(n, k) option remember;
coeff(series(t(k), x, n+1), x, n) *n!
end:
a:= n-> A(2*n,n) -A(2*n,n-1):
seq(a(n), n=0..14);
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t[k_] := t[k] = If[k < 0, 0, Exp[Sum[x^m/m!*t[k-m], {m, 1, k}]]];
A[n_, k_] := A[n, k] = SeriesCoefficient[t[k], {x, 0, n}]*n!;
a[n_] := A[2n, n] - A[2n, n-1];
Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Jun 27 2022, after Alois P. Heinz *)
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