A115593 -id:A115593 - cp's OEIS frontend

A073075 Generating function satisfies A(x) = exp(2A(x)x + 2A(x^3)x^3/3 + 2A(x^5)x^5/5 + 2A(x^7)x^7/7 +...).

1, 2, 6, 22, 86, 358, 1554, 6950, 31822, 148434, 702802, 3369046, 16319050, 79749294, 392711090, 1946732854, 9706813790, 48651303118, 244972282734, 1238621756174, 6286144819506, 32011282859598, 163517409895602, 837631563577814, 4301996341244810

Offset: 0

Paul D. Hanna, Aug 17 2002

Which kind of trees is counted by this sequence (see formulas)? - Joerg Arndt, Mar 04 2015

spec := [S,{B=Set(S),C=PowerSet(S),S=Prod(Z,B,C)},unlabeled]: seq(combstruct[count](spec,size=n), n=1..20); # Vladeta Jovovic, Feb 10 2005

m = 23; A[_] = 0;
Do[A[x_] = Exp[Sum[2 A[x^k] x^k/k, {k, 1, m, 2}]] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 29 2019 *)

G.f.: A(x) = exp(sum_{n>=0} 2*A(x^(2n+1))*x^(2n+1)/(2n+1)), A(0)=1, where A(x) = 1 + 2x + 6x^2 + 22x^3 + 86x^4 + 358x^5 +...

Let b(n) = a(n-1) for n>=1, then sum(n>=1, b(n)*x^n ) = x * prod(n>=1, ((1+x^n)/(1-x^n))^b(n) ); compare to A000081, A004111, and A115593. - Joerg Arndt, Mar 04 2015

A248869 Satisfies Sum_{n>=0} a(n)x^n = x Product_{n>=0} (1 + x^n + x^(2*n))^a(n).

Original entry on oeis.org

0, 1, 1, 2, 3, 7, 15, 34, 79, 190, 459, 1136, 2833, 7154, 18206, 46723, 120656, 313514, 818763, 2148434, 5660790, 14972103, 39734107, 105779291, 282403830, 755921733, 2028277115, 5454368549, 14697955778, 39682793675, 107330573239, 290783511134, 789032648219

Offset: 0

Joerg Arndt, Mar 04 2015

nonn

What kind of trees are counted by this sequence (compare with A000081, A004111, A073075, and A115593)?

a(n) is the number of rooted trees of n vertices that have everywhere at most 2 siblings with the same (i.e., isomorphic) subtree below. The g.f. assembles a(n) as a root with child subtrees from among the smaller a(), but takes only 0, 1 or 2 copies of any one of them. Compare asymmetric trees A004111 g.f. which takes 0 or 1 copies. Here the x^(2*n) term allows a 2nd copy. The siblings condition is equivalent to the condition that the tree automorphisms form a 2-group, i.e., group order some power 2^k. 2 same siblings are a swap. 3 same siblings would be an element of order 3 and hence factor 3 in the group order. a(n) >= A213920 since the latter limits same size siblings, whereas here only limits same size plus structure. - Kevin Ryde, Jul 11 2019

Alois P. Heinz, Table of n, a(n) for n = 0..2213

Cf. A000081, A004111, A073075, A115593, A213920, A248870, A309352.

Column k=2 of A318757.

h:= proc(n, m, t) option remember; `if`(m=0, binomial(n+t, t),
      `if`(n=0, 0, add(h(n-1, m-j, t+1), j=1..min(2, m))))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*h(a(i), j, 0), j=0..n/i)))
    end:
a:= n-> `if`(n<2, n, b(n-1$2)):
seq(a(n), n=0..35);  # Alois P. Heinz, Sep 04 2018

h[n_, m_, t_] := h[n, m, t] = If[m == 0, Binomial[n + t, t], If[n == 0, 0, Sum[h[n - 1, m - j, t + 1], {j, 1, Min[2, m]}]]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, i - 1]* h[a[i], j, 0], {j, 0, n/i}]]];
a[n_] := If[n < 2, n, b[n - 1, n - 1]];
a /@ Range[0, 32] (* Jean-François Alcover, Oct 02 2019, after Alois P. Heinz *)

a(n) ~ c * d^n / n^(3/2), where d = 2.8458470164106425911151048..., c = 0.41694347809945986693376... . - Vaclav Kotesovec, Mar 17 2015

a(n) = A004111(n) + A318859(n). - Kevin Ryde, Jul 11 2019

A363336 G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^(3k)) x^k/k ).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 7, 9, 11, 12, 15, 18, 20, 23, 29, 33, 38, 45, 52, 60, 72, 82, 94, 111, 128, 144, 170, 196, 222, 257, 297, 335, 388, 447, 506, 580, 668, 754, 863, 990, 1119, 1273, 1460, 1647, 1871, 2138, 2417, 2733, 3118, 3517, 3975, 4522, 5102, 5747, 6529, 7361

Offset: 0

Seiichi Manyama, May 28 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000081, A115593, A363337.

Cf. A363338.

seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, subst(A, x, x^(3*k))*x^k/k)+x*O(x^n))); Vec(A);

A(x) = Sum_{k>=0} a(k) * x^k = 1/Product_{k>=0} (1-x^(3*k+1))^a(k).
A(x) * A(w*x) * A(w^2*x) = A(x^3), where w = exp(2*Pi*i/3).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k and d==1 mod 3} d * a(floor(d/3)) ) * a(n-k).

A363337 G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^(4k)) x^k/k ).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, 8, 10, 11, 12, 14, 16, 18, 19, 22, 26, 29, 31, 34, 40, 45, 48, 52, 60, 68, 74, 80, 90, 102, 111, 121, 136, 152, 166, 180, 201, 225, 245, 264, 294, 329, 360, 387, 426, 476, 521, 562, 615, 683, 750, 809, 883, 978, 1071, 1156, 1259, 1389, 1523

Offset: 0

Seiichi Manyama, May 28 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000081, A115593, A363336.

seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, subst(A, x, x^(4*k))*x^k/k)+x*O(x^n))); Vec(A);

A(x) = Sum_{k>=0} a(k) * x^k = 1/Product_{k>=0} (1-x^(4*k+1))^a(k).
A(x) * A(i*x) * A(-x) * A(i^3*x) = A(x^4), where i=sqrt(-1).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k and d==1 mod 4} d * a(floor(d/4)) ) * a(n-k).

A117356 Number of rooted trees with total weight n, where the weight of a node at height k is k (with the root considered to be at level 0).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 6, 8, 12, 16, 22, 31, 41, 56, 78, 104, 142, 194, 260, 353, 478, 641, 864, 1164, 1560, 2095, 2810, 3757, 5028, 6722, 8966, 11963, 15945, 21223, 28244, 37551, 49871, 66210, 87829, 116411, 154222, 204162, 270084, 357117, 471881, 623146

Offset: 0

Franklin T. Adams-Watters, Mar 09 2006

nonn

Equivalently, number of forests of total weight n, when the roots are considered to be at height 1; so this is the Euler transform of A117357. - Franklin T. Adams-Watters, Oct 03 2009

			a(3) = 2; there is one tree with 3 nodes at height 1 and one with 1 node at height 1 and 1 at height 2.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A000081, A115593, A117357.

g:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i g(n, n, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, May 16 2013

g[n_, i_, k_] := g[n, i, k] = If[n == 0, 1, If[i < k, 0, Sum[Binomial[g[i - k, i - k, k + 1] + j - 1, j] g[n - i j, i - 1, k], {j, 0, n/i}]]];
a[n_] := g[n, n, 1];
a /@ Range[0, 60] (* Jean-François Alcover, Nov 05 2020, after Alois P. Heinz *)

If a(n) is the equivalent of this sequence with the root node considered to be at level k, then a(n) is the Euler transform of a(n) shifted right k places. To compute N terms, take k so that (k+1)*(k+2)/2 > N, approximate a(n) by 1 if n=k, 0 otherwise and apply this rule repeatedly. Formula from Christian G. Bower.

A195865 G.f. satisfies A(x) = exp( Sum_{n>=1} (A(x^n) + A(-x^n))/2 * x^n/n ).

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 10, 12, 25, 33, 68, 91, 190, 264, 555, 780, 1649, 2365, 5021, 7274, 15518, 22727, 48646, 71784, 154162, 229094, 493346, 737215, 1591518, 2390072, 5170896, 7798020, 16903848, 25587218, 55561618, 84377881, 183509750, 279499063, 608726985, 929556155, 2027094432

Offset: 0

Paul D. Hanna, Oct 26 2011

nonn

For n>=1, a(n) is the number of rooted trees (see A000081) with n non-root nodes where non-root nodes cannot have odd out-degrees, see the note by David Callan and the example. - Joerg Arndt, Jun 28 2014

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 10*x^7 +...
Let B(x) = (A(x) + A(-x))/2 then
log(A(x)) = B(x) + B(x^2)*x^2/2 + B(x^3)*x^3/3 + B(x^4)*x^4/4 +...
The coefficients in (A(x) + A(-x))/2 begin:
[1,0,1,0,2,0,5,0,12,0,33,0,91,0,264,0,780,0,2365,0,7274,...]
from which the Euler transform generates the g.f. A(x):
A(x) = 1/((1-x)*(1-x^3)*(1-x^5)^2*(1-x^7)^5*(1-x^9)^12*(1-x^11)^33*(1-x^13)^91*...*(1-x^(2*n+1))^a(2*n)*...).
From _Joerg Arndt_, Jun 28 2014: (Start)
The a(6) = 5 rooted trees with 6 non-root nodes as described in the comment are:
:           level sequence       out-degrees (dots for zeros)
:     1:  [ 0 1 2 3 3 2 1 ]    [ 2 2 2 . . . . ]
:  O--o--o--o
:        .--o
:     .--o
:  .--o
:
:     2:  [ 0 1 2 2 2 2 1 ]    [ 2 4 . . . . . ]
:  O--o--o
:     .--o
:     .--o
:     .--o
:  .--o
:
:     3:  [ 0 1 2 2 1 2 2 ]    [ 2 2 . . 2 . . ]
:  O--o--o
:     .--o
:  .--o--o
:     .--o
:
:     4:  [ 0 1 2 2 1 1 1 ]    [ 4 2 . . . . . ]
:  O--o--o
:     .--o
:  .--o
:  .--o
:  .--o
:
:     5:  [ 0 1 1 1 1 1 1 ]    [ 6 . . . . . . ]
:  O--o
:  .--o
:  .--o
:  .--o
:  .--o
:  .--o
:
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..300 from Vincenzo Librandi)
David Callan, Even outdegree rooted trees, (7-July-2014).

Cf. A115593.

a[1] = 1;
a[n_] := a[n] = 1/(n - 1) Sum[(2 m + 1) a[2 m + 1] a[n - k (2 m + 1)], {m, 0, Floor[n/2] - 1}, {k, Floor[(n - 1)/(2 m + 1)]}];
Table[a[n], {n, 30}] (* Use offset 1 to simplify defining equation for G.f. Then apply xD_x, simplify, and equate coefficients to get above recurrence. - David Callan, Jul 07 2014 *)

{a(n)=my(A=1+x,B); for(i=1,n, B=(A+subst(A,x,-x))/2; A=exp(sum(m=1,n,subst(B,x,x^m+x*O(x^n))*x^m/m))); polcoeff(A,n)}

Euler transform of the coefficients in (A(x) + A(-x))/2.
G.f. satisfies: A(x) = Product_{n>=0} 1/(1 - x^(2*n+1))^a(2*n).
G.f. satisfies: A(x)*A(-x) = A(x^2).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k and d is odd} d * a(d-1) ) * a(n-k). - Seiichi Manyama, May 31 2023

A306768 G.f. A(x) satisfies: A(x) = x*exp(-A(-x) + A(-x^2)/2 - A(-x^3)/3 + A(-x^4)/4 - A(-x^5)/5 + ...).

Original entry on oeis.org

0, 1, 1, -1, -2, 2, 6, -5, -18, 15, 59, -54, -215, 199, 813, -744, -3135, 2890, 12394, -11538, -50017, 46806, 204893, -192451, -849681, 800974, 3560927, -3367656, -15058478, 14279426, 64171736, -60992032, -275304665, 262199050, 1188070488, -1133572891, -5153913606

Offset: 0

Ilya Gutkovskiy, Apr 14 2019

sign

			G.f.: A(x) = x + x^2 - x^3 - 2*x^4 + 2*x^5 + 6*x^6 - 5*x^7 - 18*x^8 + 15*x^9 + 59*x^10 - 54*x^11 - 215*x^12 + ...

Cf. A000081, A004111, A045648, A049075, A073075, A115593, A307365, A307366, A307538.

terms = 36; A[] = 0; Do[A[x] = x Exp[Sum[(-1)^k A[-x^k]/k, {k, 1, terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = SeriesCoefficient[x Product[1/(1 + x^k)^((-1)^k a[k]), {k, 1, n - 1}], {x, 0, n}]; a[0] = 0; Table[a[n], {n, 0, 36}]

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

Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+d)*d*a(d) ) * a(n-k+1).

A318133 G.f.: A(x) = Sum_{n>=1} a(n)x^n = xProduct_{n>=1} (1 + x^(2*n-1))^a(n).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 2, 2, 2, 3, 3, 3, 4, 3, 5, 6, 5, 6, 7, 6, 6, 9, 8, 9, 11, 11, 13, 14, 14, 16, 20, 21, 23, 28, 29, 32, 35, 36, 42, 47, 48, 54, 64, 64, 69, 80, 85, 93, 105, 113, 124, 139, 145, 161, 181, 192, 211, 236, 252, 273, 302, 324, 356, 396, 421, 462

Offset: 1

Ilya Gutkovskiy, Aug 18 2018

nonn

Cf. A004111, A115593.

a[n_] := a[n] = SeriesCoefficient[x Product[(1 + x^(2 k - 1))^a[k], {k, 1, n - 1}], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 76}]

A248870 Satisfies Sum_{n>=0} a(n)*x^n = x / Product_{n>=0} (1 - x^n/(1 - x^n))^a(n).

Original entry on oeis.org

0, 1, 1, 3, 8, 23, 62, 181, 513, 1513, 4476, 13483, 40933, 125845, 389769, 1217590, 3828775, 12115966, 38546124, 123238296, 395725493, 1275733730, 4127339091, 13396443708, 43610621823, 142354979662, 465838195260, 1527905193504, 5022061115901, 16539625666670, 54571760414658

Offset: 0

Joerg Arndt, Mar 04 2015

nonn

Which kind of trees is counted by this sequence (compare to A000081, A004111, A073075, A248869 and A115593)?

Cf. A248869.

A307538 G.f. A(x) satisfies: A(x) = xexp(2A(-x) + 2A(-x^3)/3 + 2A(-x^5)/5 + 2A(-x^7)/7 + 2A(-x^9)/9 + ...).

Original entry on oeis.org

0, 1, -2, -2, 10, 14, -86, -126, 858, 1302, -9378, -14606, 108954, 172698, -1319966, -2119118, 16489594, 26731542, -210887998, -344490170, 2747510514, 4515757426, -36336187630, -60023827438, 486540793914, 807121753178, -6582918170714, -10959656342678, 89860260268098

Offset: 0

Ilya Gutkovskiy, Apr 14 2019

sign

			G.f.: A(x) = x - 2*x^2 - 2*x^3 + 10*x^4 + 14*x^5 - 86*x^6 - 126*x^7 + 858*x^8 + 1302*x^9 - 9378*x^10 - 14606*x^11 + ...

Cf. A000081, A004111, A045648, A049075, A073075, A115593, A306768, A307365, A307366.

terms = 28; A[] = 0; Do[A[x] = x Exp[Sum[2 A[-x^(2 k - 1)]/(2 k - 1), {k, 1, terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = SeriesCoefficient[x Product[((1 + x^k)/(1 - x^k))^((-1)^k a[k]), {k, 1, n - 1}], {x, 0, n}]; a[0] = 0; Table[a[n], {n, 0, 28}]

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

Recurrence: a(n+1) = (2/n) * Sum_{k=1..n} ( Sum_{d|k, k/d odd} (-1)^d*d*a(d) ) * a(n-k+1).

cp's OEIS Frontend

A073075 Generating function satisfies A(x) = exp(2*A(x)*x + 2*A(x^3)*x^3/3 + 2*A(x^5)*x^5/5 + 2*A(x^7)*x^7/7 +...).

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A248869 Satisfies Sum_{n>=0} a(n)*x^n = x * Product_{n>=0} (1 + x^n + x^(2*n))^a(n).

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A363336 G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^(3*k)) * x^k/k ).

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A363337 G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^(4*k)) * x^k/k ).

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A117356 Number of rooted trees with total weight n, where the weight of a node at height k is k (with the root considered to be at level 0).

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A195865 G.f. satisfies A(x) = exp( Sum_{n>=1} (A(x^n) + A(-x^n))/2 * x^n/n ).

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A306768 G.f. A(x) satisfies: A(x) = x*exp(-A(-x) + A(-x^2)/2 - A(-x^3)/3 + A(-x^4)/4 - A(-x^5)/5 + ...).

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A318133 G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x*Product_{n>=1} (1 + x^(2*n-1))^a(n).

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A248870 Satisfies Sum_{n>=0} a(n)*x^n = x / Product_{n>=0} (1 - x^n/(1 - x^n))^a(n).

A073075 Generating function satisfies A(x) = exp(2A(x)x + 2A(x^3)x^3/3 + 2A(x^5)x^5/5 + 2A(x^7)x^7/7 +...).

A248869 Satisfies Sum_{n>=0} a(n)x^n = x Product_{n>=0} (1 + x^n + x^(2*n))^a(n).

A363336 G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^(3k)) x^k/k ).

A363337 G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^(4k)) x^k/k ).

A318133 G.f.: A(x) = Sum_{n>=1} a(n)x^n = xProduct_{n>=1} (1 + x^(2*n-1))^a(n).

A307538 G.f. A(x) satisfies: A(x) = xexp(2A(-x) + 2A(-x^3)/3 + 2A(-x^5)/5 + 2A(-x^7)/7 + 2A(-x^9)/9 + ...).