A127525 -id:A127525 - cp's OEIS frontend

A127524 Number of unordered rooted trees where each subtree from given node has the same number of nodes.

1, 1, 2, 3, 5, 6, 11, 12, 20, 25, 42, 43, 81, 82, 150, 192, 287, 288, 563, 564, 982, 1277, 2182, 2183, 3658, 3785, 7108, 8659, 13101, 13102, 27827, 27828, 47768, 61025, 102355, 105689, 170882, 170883, 329651, 421547, 606283, 606284, 1193038, 1193039, 2158117

Offset: 1

Franklin T. Adams-Watters, Jan 17 2007

nonn

			The tree shown below left counts, because the subtree shown on the left has 3 nodes and so does the one on the right and a similar condition holds for the subtrees. The tree shown on the right is not counted, because the subtree shown on the left has 3 nodes, while the one on the right has 4.
O..........O...O...O
|..........|....\./.
O...O...O..O.....O..
.\...\./....\....|..
.O...O......O...O..
..\./........\./...
...O..........O....

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000081, A127525.

with(numtheory):
a:= proc(n) option remember; `if`(n<2, n,
      add(binomial(a((n-1)/d)+d-1, d), d=divisors(n-1)))
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, May 16 2013

a[1] = 1; a[n_] := a[n] = DivisorSum[n-1, Binomial[a[(n-1)/#]+#-1, #]&]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Feb 25 2017 *)

a(1) = 1; a(n+1) = Sum_{d|n} C(a(n/d) + d-1, d).

A307781 a(1) = 1; a(n+1) = -Sum_{d|n} a(d)^(n/d).

Original entry on oeis.org

1, -1, 0, -1, -1, 0, 0, -1, -2, 1, -2, 1, -2, 1, -1, 1, -5, 4, -8, 7, -10, 9, -13, 12, -15, 15, -19, 26, -28, 27, -30, 29, -34, 41, -66, 66, -100, 99, -163, 170, -223, 222, -323, 322, -420, 453, -622, 621, -771, 770, -997, 1121, -1363, 1362, -1851, 1883, -2562, 3073, -3857, 3856

Offset: 1

Ilya Gutkovskiy, Apr 28 2019

sign

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A127525, A307780.

f:= proc(n) option remember; local d;
  -add(procname(d)^((n-1)/d), d = numtheory:-divisors(n-1))
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Apr 29 2019

a[n_] := a[n] = -Sum[a[d]^((n - 1)/d), {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 60}]

lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, va[n] = -sumdiv(n-1, d, va[d]^((n-1)/d));); va;} \\ Michel Marcus, Apr 30 2019

L.g.f.: log(Product_{n>=1} (1 - a(n)*x^n)^(1/n)) = Sum_{n>=1} a(n+1)*x^n/n.

A309633 G.f.: x * Sum_{k>=1} x^k / (1 - a(k)*x^k).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 5, 6, 4, 1, 7, 1, 13, 11, 6, 1, 7, 17, 7, 11, 31, 1, 45, 1, 10, 18, 6, 146, 34, 1, 9, 27, 141, 1, 261, 1, 78, 364, 8, 1, 44, 730, 537, 18, 145, 1, 255, 1281, 2203, 51, 33, 1, 2213, 1, 47, 7461, 221, 4722, 1159, 1, 85, 38, 27948, 1, 2342, 1, 36, 17060, 347, 63146, 3427, 1

Offset: 1

Ilya Gutkovskiy, Aug 10 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..14665

Cf. A028815 (positions of 1's), A087909, A127525, A309634.

a[n_] := a[n] = SeriesCoefficient[x Sum[x^k/(1 - a[k] x^k), {k, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 80}]
a[n_] := a[n] = Sum[a[d]^((n - 1)/d - 1) , {d, Divisors[n - 1]}]; a[1] = 0; a[2] = 1; Table[a[n], {n, 1, 80}]

seq(n)={my(v=vector(n)); for(n=1, #v-1, v[n+1]=sumdiv(n, d, v[d]^(n/d-1))); v} \\ Andrew Howroyd, Aug 10 2019

a(1) = 0; a(n+1) = Sum_{d|n} a(d)^(n/d-1).

A307780 a(1) = 1; a(n+1) = Sum_{d|n, n/d odd} a(d)^(n/d).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 5, 7, 8, 9, 17, 18, 19, 29, 29, 30, 58, 59, 91, 157, 158, 159, 284, 317, 318, 445, 573, 574, 1161, 1162, 1162, 1676, 1677, 2830, 4071, 4072, 4073, 8988, 12113, 12114, 20134, 20135, 22183, 32681, 32682, 32683, 57072, 73457, 90265, 114656, 122848, 122849, 169533

Offset: 1

Ilya Gutkovskiy, Apr 28 2019

nonn

Cf. A127525, A307781.

a[n_] := a[n] = Sum[Boole[OddQ[(n - 1)/d]] a[d]^((n - 1)/d), {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 55}]

L.g.f.: log(Product_{n>=1} ((1 + a(n)*x^n)/(1 - a(n)*x^n))^(1/(2*n))) = Sum_{n>=1} a(n+1)*x^n/n.

cp's OEIS Frontend

A127524 Number of unordered rooted trees where each subtree from given node has the same number of nodes.

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A307781 a(1) = 1; a(n+1) = -Sum_{d|n} a(d)^(n/d).

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A309633 G.f.: x * Sum_{k>=1} x^k / (1 - a(k)*x^k).

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A307780 a(1) = 1; a(n+1) = Sum_{d|n, n/d odd} a(d)^(n/d).

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