A144018 -id:A144018 - cp's OEIS frontend

A007562 Number of planted trees where non-root, non-leaf nodes an even distance from root are of degree 2.

1, 1, 1, 2, 3, 6, 10, 20, 36, 72, 137, 275, 541, 1098, 2208, 4521, 9240, 19084, 39451, 82113, 171240, 358794, 753460, 1587740, 3353192, 7100909, 15067924, 32044456, 68272854, 145730675, 311575140, 667221030, 1430892924, 3072925944, 6607832422, 14226665499

Offset: 1

N. J. A. Sloane

There is no planted tree on one node by definition.
Column k=2 of A144018. - Alois P. Heinz, Oct 17 2012
It appears that a(n) is also the number of locally non-intersecting unlabeled rooted trees with n nodes, where a tree is locally non-intersecting if the branches directly under of any non-leaf node have empty intersection. - Gus Wiseman, Aug 22 2018

			G.f. = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 10*x^7 + 20*x^8 + 36*x^9 + ...
From _Joerg Arndt_, Jun 23 2014: (Start)
The a(8) = 20 such trees have the following level sequences:
01:  [ 0 1 2 3 4 3 2 1 ]
02:  [ 0 1 2 3 3 3 2 1 ]
03:  [ 0 1 2 3 3 2 2 1 ]
04:  [ 0 1 2 3 3 2 1 1 ]
05:  [ 0 1 2 3 2 3 2 1 ]
06:  [ 0 1 2 3 2 2 2 1 ]
07:  [ 0 1 2 3 2 2 1 1 ]
08:  [ 0 1 2 3 2 1 2 1 ]
09:  [ 0 1 2 3 2 1 1 1 ]
10:  [ 0 1 2 2 2 2 2 1 ]
11:  [ 0 1 2 2 2 2 1 1 ]
12:  [ 0 1 2 2 2 1 2 1 ]
13:  [ 0 1 2 2 2 1 1 1 ]
14:  [ 0 1 2 2 1 2 2 1 ]
15:  [ 0 1 2 2 1 2 1 1 ]
16:  [ 0 1 2 2 1 1 1 1 ]
17:  [ 0 1 2 1 2 1 2 1 ]
18:  [ 0 1 2 1 2 1 1 1 ]
19:  [ 0 1 2 1 1 1 1 1 ]
20:  [ 0 1 1 1 1 1 1 1 ]
Successive levels change by at most 1 and the last level is 1, compare to the example in A000081.
(End)
From _Gus Wiseman_, Aug 22 2018: (Start)
The a(7) = 10 locally non-intersecting trees:
  (o(o(oo)))
  (o(oo(o)))
  (o(oooo))
  (oo(o(o)))
  (oo(ooo))
  (o(o)(oo))
  (ooo(oo))
  (oo(o)(o))
  (oooo(o))
  (oooooo)
(End)

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

Alois P. Heinz, Table of n, a(n) for n = 1..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A000081, A000837, A004111, A007560, A144018, A289509, A316470, A316473, A316475.

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; if n=0 then 1 else (add(d*p(d), d=divisors(n)) +add(add(d*p(d), d= divisors(j)) *b(n-j), j=1..n-1))/n fi end end: b:= etr(a): a:= n-> `if`(n<=1, n, b(n-2)): seq(a(n), n=1..40);  # Alois P. Heinz, Sep 06 2008

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, (Sum[ Sum[ d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}] + Sum[ d*p[d], {d, Divisors[n]}])/n]; b]; b = etr[a]; a[n_] := If[n <= 1, n, b[n-2]]; Table[a[n], {n, 1, 36}] (* Jean-François Alcover, Aug 01 2013, after Alois P. Heinz *)
purt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]],Intersection@@#=={}&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[purt[n]],{n,10}] (* Gus Wiseman, Aug 22 2018 *)

{a(n) = local(A); if( n<2, n>0, A = x / (1 - x) + O(x^n); for(k=2, n-2, A /= (1 - x^k + O(x^n))^polcoeff(A, k-1)); polcoeff(A, n-1))}; /* Michael Somos, Oct 06 2003 */

Shifts left 2 places under Euler transform.
G.f.: x + x^2 / (Product_{k>0} (1 - x^k)^a(k)). - Michael Somos, Oct 06 2003
a(n) ~ c * d^n / n^(3/2), where d = 2.246066877341161662499621547921... and  c = 0.68490297576105466417608032... . - Vaclav Kotesovec, Jun 23 2014
G.f. A(x) satisfies: A(x) = x + x^2 * exp(A(x) + A(x^2)/2 + A(x^3)/3 + A(x^4)/4 + ...). - Ilya Gutkovskiy, Jun 11 2021

Better description from Christian G. Bower, May 15 1998

A316074 Sequence a_k of column k shifts left k places under Weigh transform and equals signum(n) for n=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 6, 2, 2, 1, 1, 1, 12, 4, 2, 2, 1, 1, 1, 25, 6, 3, 2, 2, 1, 1, 1, 52, 10, 5, 3, 2, 2, 1, 1, 1, 113, 17, 7, 4, 3, 2, 2, 1, 1, 1, 247, 29, 10, 6, 4, 3, 2, 2, 1, 1, 1, 548, 51, 17, 8, 5, 4, 3, 2, 2, 1, 1, 1, 1226, 89, 26, 12, 7, 5, 4, 3, 2, 2, 1, 1, 1

Offset: 1

Alois P. Heinz, Jun 23 2018

			Triangle T(n,k) begins:
    1;
    1,  1;
    1,  1, 1;
    2,  1, 1, 1;
    3,  2, 1, 1, 1;
    6,  2, 2, 1, 1, 1;
   12,  4, 2, 2, 1, 1, 1;
   25,  6, 3, 2, 2, 1, 1, 1;
   52, 10, 5, 3, 2, 2, 1, 1, 1;
  113, 17, 7, 4, 3, 2, 2, 1, 1, 1;

Alois P. Heinz, Rows n = 1..200, flattened
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Columns k=1-10 give: A004111, A007560, A316075, A316076, A316077, A316078, A316079, A316080, A316081, A316082.
T(2n,n) gives A000009.
Cf. A057427, A144018, A316101.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(T(i, k), j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
T:= (n, k)-> `if`(n

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[T[i, k], j]*b[n - i*j, i - 1, k], {j, 0, n/i}]]];
T[n_, k_] := If[n < k, Sign[n], b[n - k, n - k, k]];
Table[T[n, k], {n, 1, 16}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 10 2018, after Alois P. Heinz *)

A218020 Shifts 3 places left under Euler transform with a(0)=0 and a(n)=1 for n < 3.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 3, 5, 8, 14, 23, 40, 69, 121, 212, 378, 672, 1208, 2177, 3946, 7173, 13104, 23995, 44103, 81261, 150149, 278054, 516141, 959952, 1788950, 3339656, 6245177, 11696510, 21938857, 41206395, 77496891, 145926374, 275098857, 519181163, 980848600

Offset: 0

Alois P. Heinz, Oct 18 2012

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Column k=3 of A144018.

Cf. A316075.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1,
      (add(add(d*a(d), d=divisors(j)) *b(n-j), j=1..n))/n)
    end:
a:= n-> `if`(n<3, signum(n), b(n-3)):
seq(a(n), n=0..40);

b[n_] := b[n] = If[n == 0, 1, (Sum[Sum[d*a[d], {d, Divisors[j]}]*b[n - j], {j, 1, n }])/n]; a[0] = 0; a[1] = a[2] = 1; a[n_] := b[n - 3]; Table[a[n], {n, 0, 39}] (* Jean-François Alcover, Aug 01 2013, after Alois P. Heinz *)

a(n) ~ c * d^n / n^(3/2), where d = 1.964293016979213611214370656... and c = 0.8776048696248050091050307... . - Vaclav Kotesovec, Jun 23 2014

G.f.: x + x^2 + x^3 / Product_{n>=1} (1 - x^n)^a(n). - Ilya Gutkovskiy, May 08 2019

A218021 Shifts 4 places left under Euler transform with a(0)=0 and a(n)=1 for n < 4.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 3, 5, 7, 12, 18, 30, 47, 78, 125, 209, 341, 571, 946, 1592, 2663, 4503, 7594, 12898, 21891, 37334, 63691, 109039, 186816, 320913, 551829, 950842, 1640149, 2833866, 4901658, 8490019, 14720477, 25553525, 44401638, 77232183, 134457819

Offset: 0

Alois P. Heinz, Oct 18 2012

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Column k=4 of A144018.

Cf. A316076.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1,
      (add(add(d*a(d), d= divisors(j)) *b(n-j), j=1..n))/n)
    end:
a:= n-> `if`(n<4, signum(n), b(n-4)):
seq(a(n), n=0..45);

b[n_] := b[n] = If[n == 0, 1, (Sum[Sum[d*a[d], {d, Divisors[j]}]*b[n - j], {j, 1, n }])/n]; a[n_] := If[n < 4, Sign[n], b[n - 4]]; Table[a[n], {n, 0, 41}] (* Jean-François Alcover, Aug 01 2013, after Alois P. Heinz *)

a(n) ~ c * d^n / n^(3/2), where d = 1.8065918193702780027972... and c = 1.041173202249532389463... . - Vaclav Kotesovec, Jun 23 2014

G.f.: x + x^2 + x^3 + x^4 / Product_{n>=1} (1 - x^n)^a(n). - Ilya Gutkovskiy, May 08 2019

A218022 Shifts 5 places left under Euler transform with a(0)=0 and a(n)=1 for n<5.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 3, 5, 7, 11, 16, 25, 38, 59, 91, 143, 223, 352, 555, 881, 1399, 2234, 3569, 5726, 9197, 14816, 23901, 38650, 62583, 101535, 164948, 268398, 437268, 713379, 1165156, 1905348, 3119012, 5111199, 8383837, 13765016, 22619804, 37202634

Offset: 0

Alois P. Heinz, Oct 18 2012

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Column k=5 of A144018.

Cf. A316077.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1,
      (add(add(d*a(d), d= divisors(j)) *b(n-j), j=1..n))/n)
    end:
a:= n-> `if`(n<5, signum(n), b(n-5)):
seq(a(n), n=0..45);

G.f.: x + x^2 + x^3 + x^4 + x^5 / Product_{n>=1} (1 - x^n)^a(n). - Ilya Gutkovskiy, May 08 2019

A218023 Shifts 6 places left under Euler transform with a(0)=0 and a(n)=1 for n<6.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 7, 11, 15, 23, 33, 50, 73, 112, 166, 254, 383, 587, 891, 1371, 2095, 3232, 4970, 7689, 11878, 18435, 28589, 44486, 69225, 107985, 168510, 263473, 412172, 645798, 1012516, 1589480, 2496956, 3926743, 6179504, 9733649, 15342313

Offset: 0

Alois P. Heinz, Oct 18 2012

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Column k=6 of A144018.

Cf. A316078.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1,
      (add(add(d*a(d), d= divisors(j)) *b(n-j), j=1..n))/n)
    end:
a:= n-> `if`(n<6, signum(n), b(n-6)):
seq(a(n), n=0..50);

G.f.: x + x^2 + x^3 + x^4 + x^5 + x^6 / Product_{n>=1} (1 - x^n)^a(n). - Ilya Gutkovskiy, May 08 2019

A218024 Shifts 7 places left under Euler transform with a(0)=0 and a(n)=1 for n<7.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 7, 11, 15, 22, 31, 45, 64, 94, 136, 200, 294, 435, 643, 956, 1420, 2117, 3157, 4721, 7064, 10597, 15909, 23933, 36038, 54356, 82059, 124056, 187707, 284351, 431114, 654288, 993780, 1510785, 2298471, 3499653, 5332313, 8130576

Offset: 0

Alois P. Heinz, Oct 18 2012

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Column k=7 of A144018.

Cf. A316079.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1,
      (add(add(d*a(d), d=divisors(j))*b(n-j), j=1..n))/n)
    end:
a:= n-> `if`(n<7, signum(n), b(n-7)):
seq(a(n), n=0..50);

G.f.: x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 / Product_{n>=1} (1 - x^n)^a(n). - Ilya Gutkovskiy, May 08 2019

A218025 Shifts 8 places left under Euler transform with a(0)=0 and a(n)=1 for n<8.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 43, 59, 85, 118, 170, 241, 349, 500, 728, 1051, 1534, 2226, 3252, 4735, 6929, 10117, 14829, 21715, 31893, 46828, 68920, 101442, 149589, 220650, 325939, 481659, 712605, 1054747, 1562670, 2316296, 3436200

Offset: 0

Alois P. Heinz, Oct 18 2012

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Column k=8 of A144018.

Cf. A316080.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1,
      (add(add(d*a(d), d=divisors(j))*b(n-j), j=1..n))/n)
    end:
a:= n-> `if`(n<8, signum(n), b(n-8)):
seq(a(n), n=0..50);

G.f.: x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 / Product_{n>=1} (1 - x^n)^a(n). - Ilya Gutkovskiy, May 08 2019

A218026 Shifts 9 places left under Euler transform with a(0)=0 and a(n)=1 for n<9.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 57, 80, 109, 152, 211, 296, 415, 588, 832, 1185, 1689, 2413, 3449, 4940, 7073, 10141, 14544, 20880, 29991, 43131, 62064, 89417, 128925, 186090, 268808, 388677, 562381, 814393, 1180070, 1711131

Offset: 0

Alois P. Heinz, Oct 18 2012

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Column k=9 of A144018.

Cf. A316081.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1,
      (add(add(d*a(d), d=divisors(j)) *b(n-j), j=1..n))/n)
    end:
a:= n-> `if`(n<9, signum(n), b(n-9)):
seq(a(n), n=0..60);

G.f.: x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 / Product_{n>=1} (1 - x^n)^a(n). - Ilya Gutkovskiy, May 08 2019

A218027 Shifts 10 places left under Euler transform with a(0)=0 and a(n)=1 for n<10.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 78, 104, 143, 193, 266, 362, 503, 693, 969, 1349, 1896, 2656, 3746, 5267, 7436, 10476, 14798, 20869, 29491, 41635, 58878, 83234, 117841, 166851, 236568, 335526, 476451, 676868, 962566

Offset: 0

Alois P. Heinz, Oct 18 2012

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Column k=10 of A144018.

Cf. A316082.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1,
      (add(add(d*a(d), d=divisors(j)) *b(n-j), j=1..n))/n)
    end:
a:= n-> `if`(n<10, signum(n), b(n-10)):
seq(a(n), n=0..60);

G.f.: x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 / Product_{n>=1} (1 - x^n)^a(n). - Ilya Gutkovskiy, May 08 2019

cp's OEIS Frontend

A007562 Number of planted trees where non-root, non-leaf nodes an even distance from root are of degree 2.

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A316074 Sequence a_k of column k shifts left k places under Weigh transform and equals signum(n) for n=1, 1<=k<=n, read by rows.

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A218020 Shifts 3 places left under Euler transform with a(0)=0 and a(n)=1 for n < 3.

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A218021 Shifts 4 places left under Euler transform with a(0)=0 and a(n)=1 for n < 4.

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A218022 Shifts 5 places left under Euler transform with a(0)=0 and a(n)=1 for n<5.

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A218023 Shifts 6 places left under Euler transform with a(0)=0 and a(n)=1 for n<6.

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A218024 Shifts 7 places left under Euler transform with a(0)=0 and a(n)=1 for n<7.

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A218025 Shifts 8 places left under Euler transform with a(0)=0 and a(n)=1 for n<8.

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