A317580 -id:A317580 - cp's OEIS frontend

A038046 Shifts left under transform T where Ta is (identity) DCONV a.

1, 1, 3, 6, 12, 17, 32, 39, 63, 81, 120, 131, 213, 226, 311, 377, 503, 520, 742, 761, 1031, 1169, 1442, 1465, 2008, 2093, 2558, 2801, 3465, 3494, 4591, 4622, 5628, 6054, 7111, 7390, 9321, 9358, 10899, 11616, 13873, 13914, 17070, 17113, 20063, 21509, 24462

Offset: 1

Christian G. Bower

Eigensequence of triangle A126988. (i.e. the sequence shifts upon multiplication from the left by triangle A126988). - Gary W. Adamson, Apr 27 2009

Number of planted achiral trees with a distinguished leaf. - Gus Wiseman, Jul 31 2018

			From _Gus Wiseman_, Jul 31 2018: (Start)
The a(5) = 12 planted achiral trees with a distinguished leaf:
  (Oooo), (oOoo), (ooOo), (oooO),
  ((O)(o)), ((o)(O)),
  ((Ooo)), ((oOo)), ((ooO)),
  (((Oo))), (((oO))),
  ((((O)))).
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A126988. - Gary W. Adamson, Apr 27 2009

Cf. A000081, A002033, A003238, A004111, A007554, A038046, A067824, A317580.

a:= proc(n) option remember; `if`(n<2, n, (m-> m*
      add(a(d)/d, d=numtheory[divisors](m)))(n-1))
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, May 09 2019

a[n_]:=If[n==1,1,Sum[d*a[(n-1)/d],{d,Divisors[n-1]}]];
Array[a,30] (* Gus Wiseman, Jul 31 2018 *)

a(1) = 1; a(n > 1) = Sum_{d|(n-1)} d * a((n-1)/d). - Gus Wiseman, Jul 31 2018

G.f. A(x) satisfies: A(x) = x * (1 + Sum_{j>=1} j*A(x^j)). - Ilya Gutkovskiy, May 09 2019

A003227 Endpoints (leaves) in rooted trees with n nodes.

Original entry on oeis.org

1, 1, 3, 8, 22, 58, 160, 434, 1204, 3341, 9363, 26308, 74376, 210823, 599832, 1710803, 4891876, 14015505, 40231632, 115669419, 333052242, 960219982, 2771707332, 8009222307, 23166563032, 67069289457, 194332834601

Offset: 1

N. J. A. Sloane

nonn

Number of unlabeled rooted trees with n nodes and a distinguished leaf. - Gus Wiseman, Jul 31 2018

			The a(4) = 8 rooted trees with a distinguished leaf are (((O))), ((Oo)), ((oO)), (O(o)), (o(O)), (Ooo), (oOo), (ooO). - _Gus Wiseman_, Jul 31 2018

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. W. Robinson and A. J. Schwenk, The distribution of trees in a large random tree, Discr. Math., 12 (1975), 359-372.
Eric Weisstein's World of Mathematics, Tree Leaf.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A000081, A003228, A004111, A038046, A055277, A317580.

urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];
Table[Sum[Length[Flatten[{t/.{}->1}]],{t,urt[n]}],{n,15}] (* Gus Wiseman, Jul 31 2018 *)

a(n) = Sum_{k=1..n} k*A055277(n, k).

Corrected and extended with formula by Christian G. Bower, May 25 2000

cp's OEIS Frontend

A038046 Shifts left under transform T where Ta is (identity) DCONV a.

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