A301462
Number of enriched r-trees of size n.
Original entry on oeis.org
1, 2, 3, 8, 23, 77, 254, 921, 3249, 12133, 44937, 172329, 654895, 2565963, 9956885, 39536964, 156047622, 626262315, 2499486155, 10129445626, 40810378668, 166475139700, 676304156461, 2775117950448, 11342074888693, 46785595997544, 192244951610575, 796245213910406
Offset: 0
The a(3) = 8 enriched r-trees: 3, (2), ((1)), ((())), (11), (1()), (()1), (()()).
Cf.
A000081,
A003238,
A004111,
A032305,
A055277,
A093637,
A127524,
A196545,
A289501,
A290689,
A300443,
A301342-
A301345,
A301364-
A301368,
A301422,
A301467,
A301469,
A301470.
-
ert[n_]:=ert[n]=1+Sum[Times@@ert/@y,{y,IntegerPartitions[n-1]}];
Array[ert,30]
-
seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x^n)), n-1)); concat([1], v)} \\ Andrew Howroyd, Aug 26 2018
A301467
Number of enriched r-trees of size n with no empty subtrees.
Original entry on oeis.org
1, 2, 4, 8, 20, 48, 136, 360, 1040, 2944, 8704, 25280, 76320, 226720, 692992, 2096640, 6470016, 19799936, 61713152, 190683520, 598033152, 1863995392, 5879859200, 18438913536, 58464724992, 184356152832, 586898946048, 1859875518464, 5941384080384, 18901502482432
Offset: 1
The a(4) = 8 enriched r-trees with no empty subtrees: 4, (3), (21), ((2)), (111), ((11)), ((1)1), (((1))).
The a(5) = 20 enriched r-trees with no empty subtrees:
5,
(4), ((3)), ((21)), (((2))), ((111)), (((11))), (((1)1)), ((((1)))),
(31), (22), (2(1)), ((2)1), ((1)2), ((11)1), ((1)(1)), (((1))1),
(211), ((1)11),
(1111).
Cf.
A000081,
A004111,
A032305,
A055277,
A093637,
A127524,
A196545,
A289501,
A300660,
A301342-
A301345,
A301364-
A301368,
A301422,
A301462,
A301469,
A301470.
-
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1)* a(i)^j, j=0..n/i)))
end:
a:= n-> `if`(n<2, n, 1+b(n-1$2)):
seq(a(n), n=1..30); # Alois P. Heinz, Jun 21 2018
-
pert[n_]:=pert[n]=If[n===1,1,1+Sum[Times@@pert/@y,{y,IntegerPartitions[n-1]}]];
Array[pert,30]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
Sum[b[n - i*j, i - 1] a[i]^j, {j, 0, n/i}]]];
a[n_] := a[n] = If[n < 2, n, 1 + b[n-1, n-1]];
Array[a, 30] (* Jean-François Alcover, May 09 2021, after Alois P. Heinz *)
-
seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, v[n] = 1 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x^n)), n-1)); v} \\ Andrew Howroyd, Aug 26 2018
A301470
Signed recurrence over enriched r-trees: a(n) = (-1)^n + Sum_y Product_{i in y} a(y) where the sum is over all integer partitions of n - 1.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 2, 3, 5, 9, 15, 27, 47, 87, 155, 288, 524, 983, 1813, 3434, 6396, 12174, 22891, 43810, 82925, 159432, 303559, 585966, 1121446, 2171341, 4172932, 8106485, 15635332, 30445899, 58925280, 115014681, 223210718, 436603718, 849480835, 1664740873
Offset: 0
Cf.
A032305,
A055277,
A093637,
A127524,
A196545,
A220418,
A273866,
A273873,
A289501,
A290261,
A290971,
A301342-
A301345,
A301364-
A301368,
A301422,
A301462,
A301467,
A301469.
-
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1)+a(i)*b(n-i, min(n-i, i))))
end:
a:= n-> `if`(n<2, 1-n, b(n-2$2)+b(n-1, n-2)):
seq(a(n), n=0..45); # Alois P. Heinz, Jun 23 2018
-
a[n_]:=a[n]=(-1)^n+Sum[Times@@a/@y,{y,IntegerPartitions[n-1]}];
Array[a,30]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1,
If[i < 1, 0, b[n, i - 1] + a[i] b[n - i, Min[n - i, i]]]];
a[n_] := If[n < 2, 1 - n, b[n - 2, n - 2] + b[n - 1, n - 2]];
a /@ Range[0, 45] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)
Showing 1-3 of 3 results.
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