A300439
Number of odd enriched p-trees of weight n (all outdegrees are odd).
Original entry on oeis.org
1, 1, 2, 2, 5, 7, 18, 29, 75, 132, 332, 651, 1580, 3268, 7961, 16966, 40709, 89851, 215461, 484064, 1159568, 2641812, 6337448, 14622880, 35051341, 81609747, 196326305, 459909847, 1107083238, 2611592457, 6299122736, 14926657167, 36069213786, 85809507332
Offset: 1
The a(6) = 7 odd enriched p-trees: 6, (411), (321), (222), ((111)21), ((211)11), (21111).
Cf.
A000009,
A027193,
A063834,
A078408,
A196545,
A273873,
A289501,
A294079,
A298118,
A299202,
A299203,
A300300,
A300301,
A300436,
A300440.
-
f[n_]:=f[n]=1+Sum[Times@@f/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&OddQ[Length[#]]&]}];
Array[f,40]
-
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*x^n)) - 1/prod(k=1, n-1, 1 + v[k]*x^k + O(x*x^n)), n)/2); v} \\ Andrew Howroyd, Aug 26 2018
A300436
Number of odd p-trees of weight n (all proper terminal subtrees have odd weight).
Original entry on oeis.org
1, 1, 1, 2, 2, 5, 5, 12, 13, 35, 37, 98, 107, 304, 336, 927, 1037, 3010, 3367, 9585, 10924, 32126, 36438, 105589, 121045, 359691, 412789, 1211214, 1398168, 4188930, 4831708, 14315544, 16636297, 50079792, 58084208, 173370663, 202101971, 611487744, 712709423
Offset: 1
The a(7) = 5 odd p-trees: ((ooo)(ooo)o), (((ooo)oo)oo), ((ooooo)oo), ((ooo)oooo), (ooooooo).
Cf.
A000009,
A027193,
A063834,
A078408,
A196545,
A279374,
A279785,
A289501,
A298118,
A299202,
A299203,
A300300,
A300301,
A300355,
A300439,
A300440.
-
b[n_]:=b[n]=If[n>1,0,1]+Sum[Times@@b/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&And@@OddQ/@#&]}];
Table[b[n],{n,40}]
A300442
Number of binary strict trees of weight n.
Original entry on oeis.org
1, 1, 1, 2, 3, 6, 10, 23, 46, 108, 231, 561, 1285, 3139, 7348, 18265, 43907, 109887, 267582, 675866, 1669909, 4238462, 10555192, 26955062, 67706032, 173591181, 438555624, 1129088048, 2869732770, 7410059898, 18911818801, 48986728672, 125562853003, 326011708368
Offset: 0
The a(5) = 6 binary strict trees: 5, (41), (32), ((31)1), ((21)2), (((21)1)1).
The a(6) = 10 binary strict trees:
6,
(51), (42),
((41)1), ((32)1), ((31)2),
(((31)1)1), (((21)2)1), (((21)1)2),
((((21)1)1)1).
Cf.
A000992,
A001190,
A063834,
A196545,
A273873,
A289501,
A292432,
A293511,
A300352,
A300440,
A300443.
-
a:= proc(n) option remember;
1+add(a(j)*a(n-j), j=1..(n-1)/2)
end:
seq(a(n), n=0..40); # Alois P. Heinz, Mar 06 2018
-
k[n_]:=k[n]=1+Sum[Times@@k/@y,{y,Select[IntegerPartitions[n],Length[#]===2&&UnsameQ@@#&]}];
Array[k,40]
(* Second program: *)
a[n_] := a[n] = 1 + Sum[a[j]*a[n - j], {j, 1, (n - 1)/2}];
a /@ Range[0, 40] (* Jean-François Alcover, May 13 2021, after Alois P. Heinz *)
-
seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + sum(k=1, (n-1)\2, v[k]*v[n-k])); concat([1], v)} \\ Andrew Howroyd, Aug 25 2018
A300797
Number of strict trees of weight 2n + 1 in which all outdegrees and all leaves are odd.
Original entry on oeis.org
1, 1, 1, 1, 2, 2, 4, 6, 11, 17, 34, 59, 118, 213, 424, 799, 1606, 3072, 6216, 12172, 24650, 48710, 99333, 198237, 405526, 815267, 1673127, 3387165, 6974702, 14179418, 29285048, 59841630, 123848399, 253927322, 526936694, 1084022437, 2253778793, 4649778115
Offset: 0
The a(7) = 6 strict trees: 15, (11 3 1), (9 5 1), (7 5 3), ((7 3 1) 3 1), ((5 3 1) 5 1).
Cf.
A000009,
A000992,
A032305,
A063834,
A078408,
A089259,
A196545,
A273873,
A279785,
A289501,
A298118,
A300301,
A300352,
A300353,
A300436,
A300439,
A300440,
A300652.
-
a[n_]:=a[n]=If[OddQ[n],1,0]+Sum[Times@@a/@ptn,{ptn,Select[IntegerPartitions[n],Length[#]>1&&OddQ[Length[#]]&&UnsameQ@@#&]}];
Table[a[n],{n,1,60,2}]
-
seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(prod(k=1, n-1, 1 + v[k]*x^(2*k-1) + O(x^(2*n))) - prod(k=1, n-1, 1 - v[k]*x^(2*k-1) + O(x^(2*n))), 2*n-1)/2); v} \\ Andrew Howroyd, Aug 26 2018
Showing 1-4 of 4 results.
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