A051491 -id:A051491 - cp's OEIS frontend

A005197 a(n) = Sum_t t*F(n,t), where F(n,t) (see A033185) is the number of rooted forests with n (unlabeled) nodes and exactly t rooted trees.

1, 3, 7, 17, 39, 96, 232, 583, 1474, 3797, 9864, 25947, 68738, 183612, 493471, 1334143, 3624800, 9893860, 27113492, 74577187, 205806860, 569678759, 1581243203, 4400193551, 12273287277, 34307646762, 96093291818, 269654004899, 758014312091, 2134300171031

Offset: 1

N. J. A. Sloane. Definition clarified by N. J. A. Sloane, May 29 2012

nonn

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..600
E. M. Palmer and A. J. Schwenk, On the number of trees in a random forest, J. Combin. Theory, B 27 (1979), 109-121.

Cf. A000081, A005196, A033185.

with(numtheory):
t:= proc(n) option remember; local d, j; `if`(n<=1, n,
      (add(add(d*t(d), d=divisors(j))*t(n-j), j=1..n-1))/(n-1))
    end:
b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j) *
       binomial(t(i)+j-1, j), j=0..min(n/i, p)))))
    end:
a:= a-> add(k*b(n, n, k), k=1..n):
seq(a(n), n=1..40);  # Alois P. Heinz, Aug 20 2012

t[1] = 1; t[n_] := t[n] = Module[{d, j}, Sum[Sum[d*t[d], {d, Divisors[j]}]*t[n-j], {j, 1, n-1}]/(n-1)]; b[1, 1, 1] = 1; b[n_, i_, p_] := b[n, i, p] = If[p>n, 0, If[n == 0, 1, If[Min[i, p]<1, 0, Sum[b[n-i*j, i-1, p-j]*Binomial[t[i]+j-1, j], {j, 0, Min[n/i, p]}]]]]; a[n_] := Sum[k*b[n, n, k], {k, 1, n}]; Table[a[n] // FullSimplify, {n, 1, 30}] (* Jean-François Alcover, Mar 13 2014, after Alois P. Heinz *)

To get a(n), take row n of the triangle in A033185, multiply successive terms by 1, 2, 3, ... and sum. E.g. a(4) = 1*4+2*3+3*1+4*1 = 17.

a(n) ~ c * d^n / n^(3/2), where d = A051491 = 2.955765285..., c = 2.85007275... . - Vaclav Kotesovec, Sep 10 2014

More terms from Alois P. Heinz, Aug 20 2012

A116950 Number of functional patterns on n elements; or digraphs with maximum outdegree 1, n arrows and every point connected to an arrow.

Original entry on oeis.org

1, 2, 7, 20, 61, 174, 514, 1478, 4303, 12437, 36084, 104494, 303167, 879283, 2552803, 7413583, 21544347, 62635823, 182199853, 530228946, 1543761513, 4496523995, 13102414665, 38193626823, 111375529695, 324891970936, 948051861938, 2767336312386, 8080206646244

Offset: 0

Franklin T. Adams-Watters, Mar 29 2006

A001372 counts functional patterns from a set with n elements to itself; A000041 (partition function) counts functional patterns from a set with n elements to a disjoint set; this is the general case where the range may overlap the domain but may also include other values.

			For n=2 there are the following 7 digraphs:
o-+.o-+ o->o-+ 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 = 0..2127

Cf. A000041, A001372, A002861, A000081.

nmax = 750;
A002861 = Cases[Import["https://oeis.org/A002861/b002861.txt", "Table"], {, }][[;; nmax + 2, 2]];
A000081 = Cases[Import["https://oeis.org/A000081/b000081.txt", "Table"], {, }][[;; nmax + 2, 2]];
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}]/n]; b];
b[n_] := A002861[[n]] + A000081[[n + 2]];
a = etr[b];
a[0] = 1;
a /@ Range[0, nmax](* Jean-François Alcover, Mar 13 2020 *)

Euler transform of A002861(n) + A000081(n+1).

a(n) ~ c * d^n / sqrt(n), where d = A051491 = 2.95576528565199497471481752412..., c = 3.435908969217935496995961718... . - Vaclav Kotesovec, Sep 10 2014

A209397 L.g.f.: Sum_{n>=1} a(n)x^n/n = Sum_{n>=1} x^n/n exp( Sum_{k>=1} a(k)x^(nk)/k ).

Original entry on oeis.org

1, 3, 7, 19, 46, 129, 337, 939, 2581, 7238, 20263, 57337, 162319, 461961, 1317217, 3767035, 10792400, 30983565, 89084845, 256531814, 739658815, 2135234247, 6170505666, 17849457873, 51679366171, 149750711581, 434260829464, 1260198317509, 3659410074933

Offset: 1

Paul D. Hanna, Mar 07 2012

nonn

			L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 46*x^5/5 + 129*x^6/6 +...
Let G(x) be the g.f. of A000081, then
exp(L(x)) = G(x)/x where G(x) = x*exp( Sum_{n>=1} G(x^n)/n ) begins:
G(x) = x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 20*x^6 + 48*x^7 + 115*x^8 + 286*x^9 + 719*x^10 + 1842*x^11 + 4766*x^12 + 12486*x^13 + 32973*x^14 +...

Alois P. Heinz, Table of n, a(n) for n = 1..2136 (first 500 terms from Paul D. Hanna)

Cf. A000081, A203253.

{a(n)=local(L=vector(n, i, 1)); for(i=1, n, L=Vec(deriv(sum(m=1, n, x^m/m*exp(sum(k=1, n\m, L[k]*x^(m*k)/k)+x*O(x^n)))))); L[n]}
for(n=1,30,print1(a(n),","))

a(n) = Sum_{d|n} d*A000081(d).
L.g.f.: Sum_{n>=1} -A000081(n) * log(1-x^n).
L.g.f.: log( G(x)/x ) = Sum_{n>=1} G(x^n)/n where G(x) is the g.f. of A000081, which is the number of rooted trees with n nodes.
a(n) ~ c * d^n / sqrt(n), where d = A051491 = 2.9557652856519949747148..., c = A187770 = 0.4399240125710253040409... . - Vaclav Kotesovec, Oct 30 2014

A215978 Number of simple unlabeled graphs on n nodes with connected components that are trees or cycles.

Original entry on oeis.org

1, 1, 2, 4, 8, 14, 28, 52, 104, 206, 429, 903, 1982, 4430, 10206, 23966, 57522, 140236, 347302, 870682, 2207819, 5651437, 14590703, 37948338, 99358533, 261684141, 692906575, 1843601797, 4926919859, 13220064562, 35604359531, 96218568474, 260850911485

Offset: 0

Alois P. Heinz, Aug 29 2012

nonn

			a(4) = 8:
.o-o.  .o-o.  .o-o.  .o-o.  .o-o.  .o-o.  .o-o.  .o o.
.| |.  .|  .  .|\ .  .|/ .  .|  .  .   .  .   .  .   .
.o-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 = 0..1000

Row sum of A215977.

The labeled version is A144164. The inverse Euler transform is A215981.

with(numtheory):
b:= proc(n) option remember; local d, j; `if`(n<=1, n,
      (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/(n-1))
    end:
g:= proc(n) option remember; local k; `if`(n>2, 1, 0)+ b(n)-
      (add(b(k)*b(n-k), k=0..n) -`if`(irem(n, 2)=0, b(n/2), 0))/2
    end:
p:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i)+j-1,j)*p(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> p(n, n):
seq(a(n), n=0..40);

b[n_] := b[n] = If[n <= 1, n, (Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n - j], {j, 1, n - 1}])/(n - 1)];
g[n_] := g[n] = If[n > 2, 1, 0] + b[n] - (Sum[b[k]*b[n - k], {k, 0, n}] - If[Mod[n, 2] == 0, b[n/2], 0])/2;
p[n_, i_] := p[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[g[i] + j - 1, j]*p[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := p[n, n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 21 2017, translated from Maple *)

from sympy.core.cache import cacheit
from sympy import binomial, divisors
@cacheit
def b(n): return n if n<2 else sum([sum([d*b(d) for d in divisors(j)])*b(n - j) for j in range(1, n)])//(n - 1)
@cacheit
def g(n): return (1 if n>2 else 0) + b(n) - (sum([b(k)*b(n - k) for k in range(n + 1)]) - (b(n//2) if n%2==0 else 0))//2
@cacheit
def p(n, i): return 1 if n==0 else 0 if i<1 else sum([binomial(g(i) + j - 1, j)*p(n - i*j, i - 1) for j in range(n//i + 1)])
def a(n): return p(n, n)
print([a(n) for n in range(41)]) # Indranil Ghosh, Aug 07 2017

a(n) = Sum_{k=0..n} A215977(n,k).

a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.95576528565199497471481752412... is Otter's rooted tree constant, and c = 1.085767435235426664262830616636... . - Vaclav Kotesovec, Mar 22 2017

A255170 a(n) = A087803(n) - n + 1.

Original entry on oeis.org

1, 1, 2, 5, 13, 32, 79, 193, 478, 1196, 3037, 7802, 20287, 53259, 141069, 376449, 1011295, 2732453, 7421128, 20247355, 55469186, 152524366, 420807220, 1164532203, 3231706847, 8991343356, 25075077684, 70082143952, 196268698259, 550695545855, 1547867058852

Offset: 1

Vladimir Reshetnikov, Feb 15 2015

Conjectured extension of A199812: number of distinct values taken by w^w^...^w (with n w's and parentheses inserted in all possible ways) where w is the first transfinite ordinal omega. So far all known terms of A199812 (that is, 20 of them) coincide with this sequence. It is conjectured that A199812 is actually identical to this sequence, but it remains unproved, and is computationally difficult to check for n > 20.

			a(4) = 1 - 4 + Sum_{k=1..4} A000081(k) = 1 - 4 + 1 + 1 + 2 + 4 = 5.
a(5) = 1 - 5 + Sum_{k=1..5} A000081(k) = 1 - 5 + 1 + 1 + 2 + 4 + 9 = 13.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Libor Behounek, Ordinal Calculator
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis
MathOverflow, A discussion related to this sequence
Eric Weisstein's World of Mathematics, Ordinal Number.
Eric Weisstein's World of Mathematics, Rooted Tree.

Cf. A199812 (conjectured to be identical), A087803, A000081, A174145 (2nd differences), A005348, A002845, A198683, A187770, A051491.

with(numtheory):
t:= proc(n) option remember; `if`(n<2, n, (add(add(
      d*t(d), d=divisors(j))*t(n-j), j=1..n-1))/(n-1))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0,
      add(b(n-i*j, i-1)*binomial(t(i)+j-1, j), j=0..n/i)))
    end:
a:= proc(n) option remember; `if`(n<3, 1,
      b(n-1$2) +2*a(n-1) -a(n-2))
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, Feb 17 2015

t[1] = a[1] = 1; t[n_] := t[n] = Sum[k t[k] t[n - k m]/(n-1), {k, n}, {m, (n-1)/k}]; a[n_] := a[n] = a[n-1] + t[n] - 1; Table[a[n], {n, 40}] (* Vladimir Reshetnikov, Aug 12 2016 *)

a(n) = 1 - n + Sum_{k=1..n} A000081(k).
Recurrence: a(1) = 1, a(n+1) = a(n) + A000081(n+1) - 1.
Recurrence: a(1) = a(2) = 1, a(n) = A174145(n-1) + 2*a(n-1) - a(n-2).
Asymptotics: a(n) ~ c * d^n / n^(3/2), where c = A187770 / (1 - 1 / A051491) = 0.664861... and d = A051491 = 2.955765...

Simpler definition and program in terms of A000081. - Vladimir Reshetnikov, Aug 12 2016

Renamed. - Vladimir Reshetnikov, Aug 23 2016

A339986 Decimal expansion of a constant related to the asymptotics of A339984.

Original entry on oeis.org

0, 5, 7, 8, 4, 4, 6, 7, 8, 7, 8, 4, 8, 5, 6, 0, 5, 8, 9, 2, 2, 6, 7, 2, 8, 5, 7, 4, 8, 4, 0, 9, 3, 3, 9, 2, 5, 0, 3, 1, 1, 0, 3, 9, 2, 0, 2, 3, 0, 2, 0, 3, 8, 5, 8, 8, 7, 6, 9, 3, 6, 8, 5, 9, 5, 0, 9, 2, 2, 9, 4, 3, 7, 0, 8, 3, 1, 7, 3, 8, 1, 7, 0, 2, 2, 6, 3, 0, 4, 2, 8, 8, 0, 7, 7, 5, 0, 1, 1, 2, 1, 2, 0, 6, 8, 2

Offset: 0

Vaclav Kotesovec, Dec 25 2020

			0.057844678784856058922672857484093392503110392...

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000226, A000368, A187770, A339984, A339985.

Equals lim_{n->infinity} A339984(n) * n^(3/2) / A051491^n.

A174145 Number of rooted forests with n nodes in which each component contains at least two nodes.

Original entry on oeis.org

1, 0, 1, 2, 5, 11, 28, 67, 171, 433, 1123, 2924, 7720, 20487, 54838, 147570, 399466, 1086312, 2967517, 8137552, 22395604, 61833349, 171227674, 475442129, 1323449661, 3692461865, 10324097819, 28923331940, 81179488039, 228240293289, 642744665401, 1812762839702

Offset: 0

N. J. A. Sloane, Nov 26 2010

nonn

Row sums of A174135.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000081, A051491, A187770, A255170.

with(numtheory):
t:= proc(n) option remember; local d, j; `if`(n<=1, n,
      (add(add(d*t(d), d=divisors(j))*t(n-j), j=1..n-1))/(n-1))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0,
      add(b(n-i*j, i-1)*binomial(t(i)+j-1, j), j=0..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..32);  # Alois P. Heinz, May 17 2013

t[n_] := t[n] = If[n <= 1, n, Sum[Sum[d*t[d], {d, Divisors[j]}]*t[n-j], {j, 1, n-1}]/(n-1)]; b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<2, 0, Sum[b[n-i*j, i-1]*Binomial[t[i]+j-1, j], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n] // FullSimplify, {n, 0, 32}] (* Jean-François Alcover, Mar 19 2014, after Alois P. Heinz *)
t[1] = 1; t[n_] := t[n] = Sum[k t[k] t[n - k m]/(n-1), {k, n-1}, {m, (n-1)/k}]; a[n_] := t[n+1] - t[n]; Table[a[n], {n, 0, 32}] (* Vladimir Reshetnikov, Aug 12 2016 *)

a(n) ~ c * d^n / n^(3/2), where d = A051491 = 2.9557652856519949747148..., c = 0.8603881121111431... . - Vaclav Kotesovec, Sep 10 2014
In the asymptotics above the constant c = A187770 * (A051491 - 1). - Vladimir Reshetnikov, Aug 12 2016
a(n) = A000081(n+1) - A000081(n). - Vladimir Reshetnikov, Nov 06 2015

A215930 Number of forests on unlabeled nodes with n edges and no single node trees.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 34, 71, 154, 341, 768, 1765, 4134, 9838, 23766, 58226, 144353, 361899, 916152, 2339912, 6023447, 15617254, 40752401, 106967331, 282267774, 748500921, 1993727506, 5332497586, 14316894271, 38574473086, 104273776038, 282733466684, 768809041078

Offset: 0

Alois P. Heinz, Aug 27 2012

nonn

Each forest counted by a(n) with n>0 has number of nodes from the interval [n+1,2*n] and number of trees in [1,n].
Also limiting sequence of reversed rows of A095133.
Differs from A011782 first at n=6 (32) and from A088325 at n=8 (153).

			a(0) = 1: (  ), the empty forest with 0 trees and 0 edges.
a(1) = 1: ( o-o ), 1 tree and 1 edge.                      o
a(2) = 2: ( o-o-o ), ( o-o o-o ).                          |
a(3) = 4: ( o-o-o-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 = 0..650

Cf. A000055, A005195, A011782, A051491, A088325, A095133, A105821, A136605.

with(numtheory):
b:= proc(n) option remember; local d, j; `if`(n<=1, n,
      (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/(n-1))
    end:
t:= proc(n) option remember; local k; `if` (n=0, 1, b(n)-
      (add(b(k)*b(n-k), k=0..n)-`if`(irem(n, 2)=0, b(n/2), 0))/2)
    end:
g:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(g(n-i*j, i-1, p-j)*
       binomial(t(i)+j-1, j), j=0..min(n/i, p)))))
    end:
a:= n-> g(2*n, 2*n, n):
seq(a(n), n=0..40);

nn = 30; t[x_] := Sum[a[n] x^n, {n, 1, nn}]; a[0] = 0;
a[1] = 1; sol =
SolveAlways[
  0 == Series[
    t[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}], x];
b[x_] := Sum[a[n] x^n /. sol, {n, 0, nn}]; ft =
Drop[Flatten[
   CoefficientList[Series[b[x] - (b[x]^2 - b[x^2])/2, {x, 0, nn}],
    x]], 1]; Drop[
CoefficientList[
  Series[Product[1/(1 - y ^(i - 1))^ft[[i]], {i, 2, nn}], {y, 0, nn}],
y], -1] (* Geoffrey Critzer, Nov 10 2014 *)

a(n) = A095133(2*n,n).
a(n) = A105821(2*n+1,n+1). - Alois P. Heinz, Jul 10 2013
a(n) = A136605(2*n+1,n). - Alois P. Heinz, Apr 11 2014
a(n) ~ c * d^n / n^(5/2), where d = A051491 = 2.955765285..., c = 3.36695186... . - Vaclav Kotesovec, Sep 10 2014

A244407 Number of unlabeled rooted trees with 2n nodes and maximal outdegree (branching factor) n.

Original entry on oeis.org

1, 2, 6, 17, 50, 143, 416, 1199, 3474, 10049, 29119, 84377, 244748, 710199, 2062274, 5991418, 17416401, 50652248, 147384676, 429043390, 1249508947, 3640449679, 10610613552, 30937605076, 90237313083, 263288153074, 768449666117, 2243530461067, 6552016136667

Offset: 1

Joerg Arndt and Alois P. Heinz, Jun 27 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..100

Cf. A244372, A244410, A051491, A299039.

b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
       b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
    end:
a:= n-> b(2*n-1$2, n$2)-b(2*n-1$2, n-1$2):
seq(a(n), n=1..30);

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[b[i - 1, i - 1, k, k] + j - 1, j]* b[n - i*j, i - 1, t - j, k], {j, 0, Min[t, n/i]}]] // FullSimplify] ; a[n_] := b[2*n - 1, 2 n - 1, n, n] - b[2*n - 1, 2 n - 1, n - 1, n - 1]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 06 2015, after Maple *)

a(n) = A244372(2n,n).

a(n) ~ c * d^n / sqrt(n), where d = 2.955765285651994974714817524... is the Otter's rooted tree constant (see A051491), and c = 0.9495793... . - Vaclav Kotesovec, Jul 11 2014

A255705 Number of 2n+1-node rooted trees in which the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root equals n+1.

Original entry on oeis.org

1, 1, 3, 8, 22, 60, 167, 465, 1306, 3681, 10422, 29597, 84313, 240757, 689035, 1975753, 5675145, 16326198, 47032200, 135658367, 391733593, 1132357784, 3276330780, 9487885056, 27497891241, 79753806451, 231474005120, 672250119756, 1953523496677, 5680002466125

Offset: 0

Alois P. Heinz, Mar 02 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..800

Cf. A255636, A255704, A318754, A318758.

with(numtheory):
g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-
      `if`(d=k, 1, 0)), d=divisors(j))*g(n-j, k), j=1..n)/n)
    end:
a:= a-> g(2*n, n+1) -`if`(n=0, 0, g(2*n, n)):
seq(a(n), n=0..40);

g[n_, k_] := g[n, k] = If[n == 0, 1, Sum[DivisorSum[j, #*(g[# - 1, k] - If[# == k, 1, 0]) &]*g[n - j, k], {j, 1, n}]/n];
a[n_] :=  g[2n, n+1] - If[n == 0, 0, g[2n, n]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 24 2017, translated from Maple *)

a(n) = A255704(2*n+1,n+1).
a(n) ~ c * d^n / sqrt(n), where d = A051491 = 2.955765285651994974714817524... and c = 0.70755335886284109851526791506579... . - Vaclav Kotesovec, Feb 28 2016
a(n) = A318754(2n+2,n+1) = A318758(2n+2,n+1). - Alois P. Heinz, Sep 02 2018

cp's OEIS Frontend

A005197 a(n) = Sum_t t*F(n,t), where F(n,t) (see A033185) is the number of rooted forests with n (unlabeled) nodes and exactly t rooted trees.

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A116950 Number of functional patterns on n elements; or digraphs with maximum outdegree 1, n arrows and every point connected to an arrow.

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A209397 L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} x^n/n * exp( Sum_{k>=1} a(k)*x^(n*k)/k ).

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PARI

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A215978 Number of simple unlabeled graphs on n nodes with connected components that are trees or cycles.

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A255170 a(n) = A087803(n) - n + 1.

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A339986 Decimal expansion of a constant related to the asymptotics of A339984.

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A174145 Number of rooted forests with n nodes in which each component contains at least two nodes.

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A215930 Number of forests on unlabeled nodes with n edges and no single node trees.

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A209397 L.g.f.: Sum_{n>=1} a(n)x^n/n = Sum_{n>=1} x^n/n exp( Sum_{k>=1} a(k)x^(nk)/k ).