A187770 -id:A187770 - cp's OEIS frontend

A000343 5th power of rooted tree enumerator; number of linear forests of 5 rooted trees.

1, 5, 20, 70, 230, 721, 2200, 6575, 19385, 56575, 163952, 472645, 1357550, 3888820, 11119325, 31753269, 90603650, 258401245, 736796675, 2100818555, 5990757124, 17087376630, 48753542665, 139155765455, 397356692275, 1135163887190, 3244482184720, 9277856948255

Offset: 5

N. J. A. Sloane

nonn

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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 = 5..750
Index entries for sequences related to rooted trees

Column 5 of A339067.

Cf. A000081, A000106, A000242, A000300, A000395.

b:= proc(n) option remember; if n<=1 then n else add(k*b(k)* s(n-1, k), k=1..n-1)/(n-1) fi end: s:= proc(n,k) option remember; add(b(n+1-j*k), j=1..iquo(n,k)) end: B:= proc(n) option remember; add(b(k)*x^k, k=1..n) end: a:= n-> coeff(series(B(n-4)^5, x=0, n+1), x,n): seq(a(n), n=5..29); # Alois P. Heinz, Aug 21 2008

b[n_] := b[n] = If[n <= 1, n, Sum[k*b[k]*s[n-1, k], {k, 1, n-1}]/(n-1)]; s[n_, k_] := s[n, k] = Sum[b[n+1-j*k], {j, 1, Quotient[n, k]}]; B[n_] := B[n] = Sum[b[k]*x^k, {k, 1, n}]; a[n_] := Coefficient[Series[B[n-4]^5, {x, 0, n+1}], x, n]; Table[a[n], {n, 5, 32}] (* Jean-François Alcover, Mar 05 2014, after Alois P. Heinz *)

G.f.: B(x)^5 where B(x) is g.f. of A000081.

a(n) ~ 5 * A187770 * A051491^n / n^(3/2). - Vaclav Kotesovec, Jan 03 2021

More terms from Christian G. Bower, Nov 15 1999

A000395 6th power of rooted tree enumerator; number of linear forests of 6 rooted trees.

Original entry on oeis.org

1, 6, 27, 104, 369, 1236, 3989, 12522, 38535, 116808, 350064, 1039896, 3068145, 9004182, 26314773, 76652582, 222705603, 645731148, 1869303857, 5404655358, 15611296146, 45060069406, 129989169909, 374843799786, 1080624405287

Offset: 6

N. J. A. Sloane

nonn

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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 = 6..1000
Index entries for sequences related to rooted trees

Column 6 of A339067.

Cf. A000081, A000106, A000242, A000300, A000343.

b:= proc(n) option remember; if n<=1 then n else add(k*b(k)* s(n-1, k), k=1..n-1)/(n-1) fi end: s:= proc(n,k) option remember; add(b(n+1-j*k), j=1..iquo(n,k)) end: B:= proc(n) option remember; add(b(k)*x^k, k=1..n) end: a:= n-> coeff(series(B(n-5)^6, x=0, n+1), x,n): seq(a(n), n=6..30);  # Alois P. Heinz, Aug 21 2008

b[n_] := b[n] = If[n <= 1, n, Sum[k*b[k]*s[n-1, k], {k, 1, n-1}]/(n-1)]; s[n_, k_] := s[n, k] = Sum[b[n+1-j*k], {j, 1, Quotient[n, k]}]; B[n_] := B[n] = Sum[b[k]*x^k, {k, 1, n}]; a[n_] := SeriesCoefficient[B[n-5]^6, {x, 0, n}]; Table[a[n], {n, 6, 30}] (* Jean-François Alcover, Oct 13 2014, after Alois P. Heinz *)

G.f.: B(x)^6 where B(x) is g.f. of A000081.

a(n) ~ 6 * A187770 * A051491^n / n^(3/2). - Vaclav Kotesovec, Jan 03 2021

More terms from Christian G. Bower, Nov 15 1999

A000368 Number of connected graphs with one cycle of length 4.

Original entry on oeis.org

1, 1, 4, 9, 28, 71, 202, 542, 1507, 4114, 11381, 31349, 86845, 240567, 668553, 1860361, 5188767, 14495502, 40572216, 113743293, 319405695, 898288484, 2530058013, 7135848125, 20152898513, 56986883801

Offset: 4

N. J. A. Sloane

nonn

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973, page 69.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 4..2000 (terms 4..43 from Sean A. Irvine, 44..200 from Washington Bomfim)
Washington Bomfim, Illustration of initial terms

Column k=4 of A217781.
Cf. A000081, A000226, A001429, A005703.
Second diagonal of A058879.

Needs["Combinatorica`"]; nn = 30; s[n_, k_] := s[n, k] = a[n + 1 - k] + If[n < 2 k, 0, s[n - k, k]]; a[1] = 1; a[n_] := a[n] = Sum[a[i] s[n - 1, i] i, {i, 1, n - 1}]/(n - 1); rt = Table[a[i], {i, 1, nn}]; Take[CoefficientList[CycleIndex[DihedralGroup[4], s] /. Table[s[j] -> Table[Sum[rt[[i]] x^(k*i), {i, 1, nn}], {k, 1, nn}][[j]], {j, 1, nn}], x], {5, nn}]  (* Geoffrey Critzer, Oct 12 2012, after code given by Robert A. Russell in A000081 *)
A000081 = Rest[Cases[ Import["https://oeis.org/A000081/b000081.txt", "Table"], {, }][[All, 2]]]; max = 30; g81 = Sum[A000081[[k]]*x^k, {k, 1, max}]; g81x2 = Sum[A000081[[k]]*x^(2 k), {k, 1, max}]; g81x4 = Sum[A000081[[k]]*x^(4 k), {k, 1, max}]; Drop[CoefficientList[ Series[(2*g81x4 + 3*g81x2^2 + 2*g81^2*g81x2 + g81^4)/8, {x, 0, max}], x], 4] (* Vaclav Kotesovec, Dec 25 2020 *)

g(Q)={my(V=Vec(Q),D=Set(V),d=#D); if(d==4,return(3*f[D[1]]*f[D[2]]*f[D[3]]*f[D[4]]));
if(d==1, return((f[D[1]]^4+2*f[D[1]]^3+3*f[D[1]]^2+2*f[D[1]])/8));
my(k=1, m = #select(x->x == D[k],V), t); while(m==1, k++; m = #select(x->x == D[k], V)); t = D[1]; D[1] = D[k]; D[k] = t;
if(d == 3, return( f[D[1]] * f[D[2]] * f[D[3]] * (3 * f[D[1]] + 1)/2 ) );
if(m==3, return(f[D[1]]^2 * f[D[2]] * (f[D[1]] + 1)/2));
((3*f[D[2]]^2 + f[D[2]])*f[D[1]]^2 + (f[D[2]]^2 + 3*f[D[2]])*f[D[1]])/4 };
seq(max_n) = { my(s, a = vector(max_n), U); f = vector(max_n); f[1] = 1;
for(j=1, max_n - 1, if(j%100==0,print(j)); f[j+1] = 1/j * sum(k=1, j, sumdiv(k,d, d * f[d]) * f[j-k+1]));
for(n=4, max_n, s=0; forpart(Q = n, if( (Q[4] > Q[3]) && (Q[3]-1 > Q[2]),
      U = U / (f[Q[4] + 1] * f[Q[3] - 1]) * f[Q[4]] * f[Q[3]],  U = g(Q)); s += U,
[1,n],[4,4]); a[n] = s; if(n % 100 == 0, print(n": " s))); a[4..max_n] };
\\ Washington Bomfim, Jul 19 2012 and Dec 22 2020

From Washington Bomfim, Jul 19 2012 and Dec 22 2020: (Start)
a(n) = Sum_{P}( g(Q) ), where P is the set of the partitions Q of n with 4 parts, Q with distinct parts D[1]..D[d], D[1] the part of maximum multiplicity m in Q, f(n) = A000081(n), and g(Q) given by,
       | 3 * f(D[1]) * f(D[2]) * f(D[3]) * f(D[4]),              if d = 4,
       | (f(D[1])^4 + 2*f(D[1])^3 + 3*f(D[1])^2 + 2*f(D[1]))/8,  if d = 1,
g(Q) = | f(D[1]) * f(D[2]) * f(D[3]) * (3 * f(D[1]) + 1)/2,      if d = 3,
       | ((3*f(D[2])^2+f(D[2]))*f(D[1])^2+(f(D[2])^2+3*f(D[2]))*f(D[1]))/4,
       |                                                   if d=2, and m=2,
       | f(D[1])^2 * f(D[2]) * (f(D[1]) + 1)/2,            if d=2, and m=3.
(End)
G.f.: (2*t(x^4) + 3*t(x^2)^2 + 2*t(x)^2*t(x^2) + t(x)^4)/8 where t(x) is the g.f. of A000081. - Andrew Howroyd, Dec 03 2020
a(n) ~ (A187770 + A339986) * A051491^n / (2 * n^(3/2)). - Vaclav Kotesovec, Dec 25 2020

More terms from Vladeta Jovovic, Apr 20 2000
Definition improved by Franklin T. Adams-Watters, May 16 2006
More terms from Sean A. Irvine, Nov 14 2010

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

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

A299039 Number of rooted trees with 2n nodes where each node has at most n children.

Original entry on oeis.org

1, 1, 3, 17, 106, 693, 4690, 32754, 234746, 1719325, 12820920, 97039824, 743680508, 5759507657, 45006692668, 354425763797, 2809931206626, 22409524536076, 179655903886571, 1447023307374888, 11703779855021636, 95020085240320710, 774088021528328920

Offset: 0

Alois P. Heinz, Feb 01 2018

nonn

			a(2) = 3:
   o     o       o
   |     |      / \
   o     o     o   o
   |    / \    |
   o   o   o   o
   |
   o

Alois P. Heinz, Table of n, a(n) for n = 0..275

Cf. A051491, A100034, A187770, A244407, A299038, A299098.

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-> `if`(n=0, 1, b(2*n-1$2, n$2)):
seq(a(n), n=0..25);

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]}]]];
a[n_] := If[n == 0, 1, b[2n - 1, 2n - 1, n, n]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 04 2018, from Maple *)

a(n) = A299038(2n,n).

a(n) ~ c * d^n / n^(3/2), where d = A051491^2 = 8.736548423865419449938118272879... and c = A187770 / 2^(3/2) = 0.155536626247883986039760097126... - Vaclav Kotesovec, Feb 02 2018, updated Mar 17 2024

A029852 Number of connected functions on n points with a loop of length 3.

Original entry on oeis.org

1, 1, 3, 9, 23, 62, 169, 451, 1217, 3291, 8916, 24243, 66155, 181053, 497134, 1369064, 3780942, 10469573, 29063361, 80867990, 225508124, 630145449, 1764240907, 4948365051, 13902893423, 39124094362, 110265280739, 311208414556, 879523722747, 2488832434859

Offset: 3

Christian G. Bower

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Column 3 of A339428.

Cf. A000081.

nn = 20; f[x_] := Sum[a[n] x^n, {n, 0, nn}]; sol =
SolveAlways[
  0 == Series[
    f[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}],
  x]; b = Flatten[Table[a[n], {n, 1, nn}] /. sol]; CoefficientList[
Series[CycleIndex[CyclicGroup[3], s] /.
   Table[s[i] -> Sum[b[[k]] x^(k*i), {k, 1, nn}], {i, 1, 3}], {x, 0,
nn}], x] (* Geoffrey Critzer, Aug 08 2013 *)

G.f.: A(x) = ( B(x)^3 + 2*B(x^3) )/3 where B(x) is o.g.f. for A000081. - Geoffrey Critzer, Aug 09 2013

a(n) ~ A187770 * A051491^n / n^(3/2). - Vaclav Kotesovec, Dec 25 2020

A029853 Number of connected functions on n points with a loop of length 4.

Original entry on oeis.org

1, 1, 4, 11, 35, 97, 282, 792, 2243, 6275, 17602, 49206, 137713, 385208, 1078667, 3022342, 8478199, 23807190, 66932592, 188394855, 530911452, 1497892857, 4230987944, 11964356354, 33869704270, 95982410945, 272279600817, 773153124315, 2197492308752

Offset: 4

Christian G. Bower

nonn

Andrew Howroyd, Table of n, a(n) for n = 4..500

Column 4 of A339428.

nn = 20; f[x_] := Sum[a[n] x^n, {n, 0, nn}]; sol =
SolveAlways[
  0 == Series[
    f[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}],
  x]; b = Flatten[Table[a[n], {n, 1, nn}] /. sol]; CoefficientList[
Series[CycleIndex[CyclicGroup[4], s] /.
   Table[s[i] -> Sum[b[[k]] x^(k*i), {k, 1, nn}], {i, 1, 4}], {x, 0,
nn}], x] (* Geoffrey Critzer, Aug 08 2013 *)

G.f.: A(x) = ( B(x)^4 + B(x^2)^2 + 2*B(x^4) )/4 where B(x) is the o.g.f. for A000081. - Geoffrey Critzer, Aug 09 2013

a(n) ~ A187770 * A051491^n / n^(3/2). - Vaclav Kotesovec, Dec 25 2020

cp's OEIS Frontend

A000343 5th power of rooted tree enumerator; number of linear forests of 5 rooted trees.

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A000395 6th power of rooted tree enumerator; number of linear forests of 6 rooted trees.

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Maple

Mathematica

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A000368 Number of connected graphs with one cycle of length 4.

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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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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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A299039 Number of rooted trees with 2n nodes where each node has at most n children.

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 ).