A051491 -id:A051491 - cp's OEIS frontend

A000226 Number of n-node unlabeled connected graphs with one cycle of length 3.

1, 1, 3, 7, 18, 44, 117, 299, 793, 2095, 5607, 15047, 40708, 110499, 301541, 825784, 2270211, 6260800, 17319689, 48042494, 133606943, 372430476, 1040426154, 2912415527, 8167992598, 22947778342, 64577555147, 182009003773, 513729375064, 1452007713130

Offset: 3

N. J. A. Sloane

Number of rooted trees on n+1 nodes where root has degree 3. - Christian G. Bower
Third column of A033185. - Michael Somos, Aug 20 2018
From Washington Bomfim, Dec 22 2020: (Start)
Number of forests of 3 rooted trees with a total of n nodes.
Number of unicyclic graphs with a cycle of length 3 and a total of n nodes.
(End)

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 = 3..1000 (first 198 terms from Vincenzo Librandi)
Sergei Abramovich, Partitions of integers, Ferrers-Young diagrams, and representational efficacy of spreadsheet modeling, Spreadsheets in Education (eJSiE): Vol. 5: Iss. 2, Article 1, p. 5
Washington Bomfim, Illustration of initial terms
F. Ruskey, Combinatorial Generation, Eq.(4.27), 2003
Eric Weisstein's World of Mathematics, Rooted Tree
Index entries for sequences related to rooted trees

Column 3 of A033185 and A217781.
Cf. A000081, A001429.
For n >= 3 a(n) = A217781(n, 3) = A058879(n, n-2) = A033185(n, 3).

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; unapply(add(b(k)*x^k, k=1..n),x) end: a:= n-> coeff(series((B(n-2)(x)^3+ 3*B(n-2)(x)* B(n-2)(x^2)+ 2*B(n-2)(x^3))/6, x=0, n+1), x,n): seq(a(n), n=3..40); # Alois P. Heinz, Aug 21 2008

terms = 30; r[] = 0; Do[r[x] = x *Exp[Sum[r[x^k]/k, {k, 1, j}]] + O[x]^j // Normal, {j, 1, terms+3}]; A[x_] = (r[x]^3 + 3*r[x]*r[x^2] + 2*r[x^3])/6 + O[x]^(terms+3); Drop[CoefficientList[A[x], x], 3] (* Jean-François Alcover, Nov 23 2011, updated Jan 11 2018 *)

seq(max_n) = {my(a = f = vector(max_n), s, D); f[1]=1;
for(j=1, max_n - 1, f[j+1] = 1/j * sum(k=1, j, sumdiv(k,d, d * f[d]) * f[j-k+1]));
for(n=3,max_n,s=0;forpart(P=n,D=Set(P);if(#D==3,s+=f[P[1]]*f[P[2]]*f[P[3]];next());
if(#D==1, s+= binomial(f[P[1]]+2, 3); next());
if(P[1] == P[2], s += binomial(f[P[1]]+1, 2) * f[P[3]],
s += binomial(f[P[2]]+1, 2) * f[P[1]]),[1,n],[3,3]); a[n] = s ); a[3..max_n] }; \\ Washington Bomfim, Dec 22 2020

G.f.: (r(x)^3+3*r(x)*r(x^2)+2*r(x^3))/6 where r(x) is g.f. for rooted trees (A000081).
a(n) = Sum_{j1+2j2+···= n} (Product_{i=1..n} binomial(A000081(i) + j_i -1, j_i)) [(4.27) of [F. Ruskey] with n replaced by n+1]. - Washington Bomfim, Dec 22 2020
a(n) ~ (A187770 + A339986) * A051491^n / (2 * n^(3/2)). - Vaclav Kotesovec, Dec 25 2020

More terms from Vladeta Jovovic, Apr 19 2000

A036364 Number of free n-ominoes with cell centers determining n-2 space (proper dimension n-2).

Original entry on oeis.org

1, 4, 11, 35, 104, 319, 951, 2862, 8516, 25369, 75167, 222529, 656961, 1937393, 5704426, 16781247, 49320800, 144866243, 425263010, 1247877578, 3660478408, 10734834603, 31475111515, 92273758477, 270486112046, 792836030163, 2323835125879, 6811162237825

Offset: 3

Robert A. Russell

Lunnon's DE(n,n-2); Lunnon's DE(n,n-1) is number of free trees.

			1 tromino in 1-space;
4 nonstraight tetrominoes in 2-space;
11 nonflat pentominoes in 3-space (chiral pairs count as one).

Vaclav Kotesovec, Table of n, a(n) for n = 3..1000 (terms 3..200 from Vincenzo Librandi)
W. F. Lunnon, Counting Multidimensional Polyominoes, Computer Journal, Vol. 18 (1975), pp. 366-67.

Cf. A000081, A000055, A036365, A171860 (fixed).

sb[ n_, k_ ] := sb[ n, k ]=b[ n+1-k, 1 ]+If[ n<2k, 0, sb[ n-k, k ] ]; b[ 1, 1 ] := 1;
b[ n_, 1 ] := b[ n, 1 ]=Sum[ b[ i, 1 ]sb[ n-1, i ]i, {i, 1, n-1} ]/(n-1);
b[ n_, k_ ] := b[ n, k ]=Sum[ b[ i, 1 ]b[ n-i, k-1 ], {i, 1, n-1} ];
Table[ b[ i, 3 ]/2+5b[ i, 4 ]/8+Sum[ b[ i, j ], {j, 5, i} ]+If[ OddQ[ i ], 0, 7b[ i/2, 2 ]/8
+If[ OddQ[ i/2 ], 0, b[ i/4, 1 ]/4 ]+Sum[ b[ i/2, j ], {j, 3, i/2} ] ]
+Sum[ b[ j, 1 ](b[ i-2j, 1 ]/2+b[ i-2j, 2 ]/4)+Sum[ If[ OddQ[ k ], b[ j,
(k-1)/2 ]b[ i-2j, 1 ], 0 ], {k, 5, i} ], {j, 1, (i-1)/2} ], {i, 3, 30} ]

G.f.: B^3(x)/2 + B(x)B(x^2)/2 + 5B^4(x)/8 + B^2(x)B(x^2)/4 + 7B^2(x^2)/8 + B(x^4)/4 + B^5(x)/(1-B(x)) + (B(x)+B(x^2))B^2(x^2)/(1-B(x^2)), where B(x) is the generating function for rooted trees with n nodes (that is, B(x) is the g.f. of sequence A000081).

a(n) ~ A340310 * A051491^n / sqrt(n). - Vaclav Kotesovec, Apr 12 2021

A051573 INVERTi transform of A000081 = [1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12486,...].

Original entry on oeis.org

1, 1, 1, 2, 3, 8, 16, 41, 98, 250, 631, 1646, 4285, 11338, 30135, 80791, 217673, 590010, 1606188, 4392219, 12055393, 33206321, 91752211, 254261363, 706465999, 1967743066, 5493195530, 15367129299, 43073007846, 120949992543, 340206026166, 958444631917

Offset: 0

N. J. A. Sloane

nonn

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

Cf. A052250, A000081.

with(numtheory):
b:= proc(n) option remember; local d, j; `if` (n<2, n,
      (add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
    end:
a:= proc(n) option remember; local i; `if`(n<0, -1,
      -add(a(n-i) *b(i+1), i=1..n+1))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 17 2013

b[n_] := b[n] = If[n < 2, n, Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}]/(n-1)]; a[n_] := a[n] = If[n < 0, -1, -Sum[a[n-i]*b[i+1], {i, 1, n+1}]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 16 2014, after Alois P. Heinz *)

a(n) ~ c * d^n / n^(3/2), where d = A051491 = 2.9557652856519949747148175241..., c = A187770 = 0.4399240125710253040409033914... . - Vaclav Kotesovec, Sep 06 2014

A086308 Decimal expansion of Otter's asymptotic constant beta for the number of unrooted trees.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jul 15 2003

A000055(n) ~ 0.5349496061 * alpha^n * n^(-5/2), where alpha = 2.95576528565199497... (see A051491). - Vaclav Kotesovec, Jan 04 2013

			0.53494960614230701455037971105206839814311651405699...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6., p. 296.

Eric Weisstein's World of Mathematics, Tree

Cf. A000055, A000081, A051491, A187770.

digits = 86; max = 250; 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[k]*s[n-1, k]*k, {k, 1, n-1}]/(n-1); A[x_] := Sum[a[k]*x^k, {k, 0, max}]; APrime[x_] := Sum[k*a[k]*x^(k-1), {k, 0, max}]; eq = Log[c] == 1 + Sum[A[c^-k]/k, {k, 2, max}]; alpha = c /. FindRoot[eq, {c, 3}, WorkingPrecision -> digits+5]; b = Sqrt[(1+Sum[APrime[alpha^-k]/alpha^k, {k, 2, max}])/(2*Pi)]; beta = 2*Pi*b^3; RealDigits[beta, 10, digits] // First (* Jean-François Alcover, Sep 24 2014 *)

Corrected and extended by Vaclav Kotesovec, Jan 04 2013

More terms from Vaclav Kotesovec, Jun 20 2013 and Dec 26 2020

A000242 3rd power of rooted tree enumerator; number of linear forests of 3 rooted trees.

Original entry on oeis.org

1, 3, 9, 25, 69, 186, 503, 1353, 3651, 9865, 26748, 72729, 198447, 543159, 1491402, 4107152, 11342826, 31408719, 87189987, 242603970, 676524372, 1890436117, 5292722721, 14845095153, 41708679697, 117372283086, 330795842217

Offset: 3

N. J. A. Sloane

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).

Column 3 of A339067.

Cf. A000081, A000106, A000300, A000343, 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-2)^3, x=0, n+1), x,n): seq(a(n), n=3..29); # Alois P. Heinz, Aug 21 2008

max = 29; 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]}]; f[x_] := Sum[ b[k]*x^k, {k, 0, max}]; Drop[ CoefficientList[ Series[f[x]^3, {x, 0, max}], x], 3] (* Jean-François Alcover, Oct 25 2011, after Alois P. Heinz *)

G.f.: B(x)^3 where B(x) is g.f. of A000081.

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

More terms from Christian G. Bower, Nov 15 1999

A000300 4th power of rooted tree enumerator: linear forests of 4 rooted trees.

Original entry on oeis.org

1, 4, 14, 44, 133, 388, 1116, 3168, 8938, 25100, 70334, 196824, 550656, 1540832, 4314190, 12089368, 33911543, 95228760, 267727154, 753579420, 2123637318, 5991571428, 16923929406, 47857425416, 135478757308, 383929643780, 1089118243128, 3092612497260

Offset: 4

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 = 4..500
Index entries for sequences related to rooted trees

Column 4 of A339067.

Cf. A000081, A000106, A000242, A000343, 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-3)^4, x=0, n+1), x,n): seq(a(n), n=4..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, n/k}]; bb[n_] := bb[n] = Sum[b[k]*x^k, {k, 1, n}]; a[n_] := Coefficient[ Series[ bb[n - 3]^4, {x, 0, n + 1}], x, n]; Table[a[n], {n, 4, 31}] (* Jean-François Alcover, Jan 25 2013, translated from Alois P. Heinz's Maple program *)

G.f.: B(x)^4 where B(x) is g.f. of A000081.

a(n) ~ 4 * A187770 * A051491^n / n^(3/2). - Vaclav Kotesovec, Jan 03 2021

More terms from Christian G. Bower, Nov 15 1999

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

Original entry on oeis.org

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

A126285 Number of partial mappings (or mapping patterns) from n points to themselves; number of partial endofunctions.

Original entry on oeis.org

1, 2, 6, 16, 45, 121, 338, 929, 2598, 7261, 20453, 57738, 163799, 465778, 1328697, 3798473, 10883314, 31237935, 89812975, 258595806, 745563123, 2152093734, 6218854285, 17988163439, 52078267380, 150899028305, 437571778542, 1269754686051, 3687025215421

Offset: 0

Christian G. Bower, Dec 25 2006 based on a suggestion from Jonathan Vos Post

nonn

If an endofunction is partial, then some points may be unmapped (or mapped to "undefined").

The labeled version is left-shifted A000169. - Franklin T. Adams-Watters, Jan 16 2007

Alois P. Heinz, Table of n, a(n) for n = 0..750
R. I. McLachlan, K. Modin, H. Munthe-Kaas, and O. Verdier, What are Butcher series, really? The story of rooted trees and numerical methods for evolution equations, arXiv preprint arXiv:1512.00906 [math.NA], 2015-2017.
N. J. A. Sloane, Transforms

Cf. A001372.

Cf. A000169, A000081, A002861.

nmax = 28;
a81[n_] := a81[n] = If[n<2, n, Sum[Sum[d*a81[d], {d, Divisors[j]}]*a81[n-j ], {j, 1, n-1}]/(n-1)];
A[n_] := A[n] = If[n<2, n, Sum[DivisorSum[j, #*A[#]&]*A[n-j], {j, 1, n-1} ]/(n-1)];
H[t_] := Sum[A[n]*t^n, {n, 0, nmax+2}];
F = 1/Product[1 - H[x^n], {n, 1, nmax+2}] + O[x]^(nmax+2);
A1372 = CoefficientList[F, x];
a[n_] := Sum[a81[k] * A1372[[n-k+2]], {k, 0, n+1}];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Aug 18 2018, after Franklin T. Adams-Watters *)

Pol. = InfinitePolynomialRing(QQ)
@cached_function
def Z(n):
    if n==0: return Pol.one()
    return sum(t[k]*Z(n-k) for k in (1..n))/n
def pmagmas(n,k=1): # number of partial k-magmas on a set of n elements up to isomorphism
    P = Z(n)
    q = 0
    coeffs = P.coefficients()
    count = 0
    for m in P.monomials():
        p = 1
        V = m.variables()
        T = cartesian_product(k*[V])
        for t in T:
            r = [Pol.varname_key(str(u))[1] for u in t]
            j = [m.degree(u) for u in t]
            D = 1
            lcm_r = lcm(r)
            for d in divisors(lcm_r):
                try: D += d*m.degrees()[-d-1]
                except: break
            p *= D^(prod(r)/lcm_r*prod(j))
        q += coeffs[count]*p
        count += 1
    return q
# Philip Turecek, Nov 27 2023

Euler transform of A002861 + A000081 = [1, 2, 4, 9, 20, 51, 125, 329, 862, 2311, ... ] + [ 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, ...] = A124682.
Convolution of left-shifted A000081 with A001372. - Franklin T. Adams-Watters, Jan 16 2007
a(n) ~ c * d^n / sqrt(n), where d = 2.95576528565... is the Otter's rooted tree constant (see A051491) and c = 1.309039781943936352117502717... - Vaclav Kotesovec, Mar 29 2014

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

cp's OEIS Frontend

A000226 Number of n-node unlabeled connected graphs with one cycle of length 3.

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A036364 Number of free n-ominoes with cell centers determining n-2 space (proper dimension n-2).

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A051573 INVERTi transform of A000081 = [1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12486,...].

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A086308 Decimal expansion of Otter's asymptotic constant beta for the number of unrooted trees.

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A000242 3rd power of rooted tree enumerator; number of linear forests of 3 rooted trees.

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A000300 4th power of rooted tree enumerator: linear forests of 4 rooted trees.

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

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