A036249 -id:A036249 - cp's OEIS frontend

A048802 Number of labeled rooted trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

1, 3, 16, 133, 1521, 22184, 393681, 8233803, 198342718, 5408091155, 164658043397, 5537255169582, 203840528337291, 8153112960102283, 352079321494938344, 16325961781591781401, 809073412162081974237, 42674870241038732398720, 2386963662244981472850709

Offset: 1

Christian G. Bower, Mar 15 1999

nonn

			G.f. = x + 3*x^2 + 16*x^3 + 133*x^4 + 1521*x^5 + 22184*x^6 + 393681*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..200
F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 861
B. R. Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.
Index entries for sequences related to rooted trees

Cf. A036249, A038052, A058863, A052807.

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[ ComposeSeries[ Series[t,{x,0,nn}],Series[Exp[x]-1 ,{x,0,nn}]],x]  (* Geoffrey Critzer, Sep 16 2012 *)

{a(n) = sum( k=1, n, stirling(n, k, 2) * k^(k - 1))}; /* Michael Somos, Jun 09 2012 */

{a(n) = n! * polcoeff( serreverse( log(1 + x*exp(-x +x*O(x^n))) ),n)}
for(n=1,30,print1(a(n),", ")) \\ Paul D. Hanna, Jan 24 2016

E.g.f.: B(exp(x)-1) where B is e.g.f. of A000169.
E.g.f.: Series_Reversion( log(1 + x*exp(-x)) ). - Paul D. Hanna, Jan 24 2016
a(n) = Sum_{k=1..n} Stirling2(n, k)*k^(k-1). - Vladeta Jovovic, Sep 17 2003
Stirling transform of A000169. - Michael Somos, Jun 09 2012
a(n) ~ sqrt(1+exp(1)) * n^(n-1) / (exp(n) * (log(1+exp(-1)))^(n-1/2)). - Vaclav Kotesovec, Feb 17 2014

A052855 Number of forests of rooted trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

Original entry on oeis.org

1, 1, 3, 8, 24, 71, 224, 710, 2318, 7659, 25703, 87153, 298574, 1031104, 3587263, 12558652, 44214807, 156438309, 555973965, 1983817178, 7104313970, 25525304569, 91986529421, 332408847422, 1204259931815, 4373027942634, 15914143511582, 58030451159889

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Euler transform of A036249 (as well as first differences thereof). - Franklin T. Adams-Watters, Feb 08 2006

Alois P. Heinz, Table of n, a(n) for n = 0..1717
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 823

First differences of A036249 and A029856.

spec := [S,{B=Sequence(Z,1 <= card),S=Set(C),C=Prod(B,S)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

max = 26; A[] = 1; Do[A[x] = Exp[Sum[A[x^k]/(1 - x^k)*x^k/k + O[x]^n, {k, 1, n}]] // Normal, {n, 1, max}]; CoefficientList[A[x] + O[x]^max, x] (* Jean-François Alcover, May 25 2018 *)

{a(n)=my(A=1+x);for(i=1,n,A=exp(sum(m=1,n,subst(A/(1-x),x,x^m+x*O(x^n))*x^m/m)));polcoeff(A,n)} /* Paul D. Hanna, Oct 26 2011 */

G.f. satisfies A(x) = exp( Sum_{n>=1} A(x^n)/(1-x^n) * x^n/n ). - Paul D. Hanna, Oct 26 2011

G.f.: A(x) = Sum_{k>=0} a(k) * x^k = 1/Product_{j>=1} Product_{k>=0} (1-x^(j+k))^a(k). - Seiichi Manyama, Jun 07 2023

More terms from Franklin T. Adams-Watters, Feb 08 2006

A029856 Number of rooted trees with 2-colored leaves.

Original entry on oeis.org

2, 2, 5, 13, 37, 108, 332, 1042, 3360, 11019, 36722, 123875, 422449, 1453553, 5040816, 17599468, 61814275, 218252584, 774226549, 2758043727, 9862357697, 35387662266, 127374191687, 459783039109, 1664042970924, 6037070913558, 21951214425140, 79981665585029

Offset: 1

Christian G. Bower

Alois P. Heinz, Table of n, a(n) for n = 1..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 768
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Essentially the same as A036249.

Cf. A000081, A029857, A038049.

A:= proc(n) option remember; if n=0 then 0 else convert(series(x+x* exp(sum(subs(x=x^i, A(n-1))/i, i=1..n-1)), x=0, n+1), polynom) fi end; a:= n-> coeff(A(n), x,n): seq(a(n), n=1..25); # Alois P. Heinz, Aug 22 2008
# second Maple program:
with(numtheory): a:= proc(n) option remember; local d,j; if n<=1 then 2*n else (add(d*a(d), d=divisors(n-1)) +add(add(d*a(d), d=divisors(j)) *a(n-j), j=1..n-2))/ (n-1) fi end: seq(a(n), n=1..25); # Alois P. Heinz, Sep 06 2008

a[n_] := a[n] = If [n <= 1, 2*n, (Sum[d*a[d], {d, Divisors[n-1]}] + Sum[Sum[d*a[d], {d, Divisors[j]}]*a[n-j], {j, 1, n-2}])/(n-1)]; Array[a, 25] (* Jean-François Alcover, Mar 13 2015, after Alois P. Heinz *)

{a(n)=local(A=x+x*O(x^n));for(i=1,n, A=x+x*exp(sum(m=1,n,subst(A,x,x^m)/m)));polcoeff(A,n,x)} \\ Paul D. Hanna, Oct 19 2005

Shifts left under Euler transform.
G.f. satisfies: A(x) = x + x*exp( Sum_{n>=1} A(x^n)/n ). - Paul D. Hanna, Oct 19 2005
a(n) ~ c * d^n / n^(3/2), where d = 3.848442876944251389076286931217197... and c = 0.48335853985605895591573724406549734... - Vaclav Kotesovec, Mar 29 2014

A036250 Number of trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

Original entry on oeis.org

1, 1, 2, 3, 7, 14, 35, 85, 231, 633, 1845, 5461, 16707, 51945, 164695, 529077, 1722279, 5664794, 18813369, 62996850, 212533226, 721792761, 2466135375, 8471967938, 29249059293, 101440962296, 353289339927, 1235154230060, 4333718587353, 15255879756033

Offset: 0

Christian G. Bower, Nov 15 1998

nonn

Also the number of non-isomorphic connected multigraphs with loops with n edges and multiset density -1, where the multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices. - Gus Wiseman, Nov 28 2018

Alois P. Heinz, Table of n, a(n) for n = 0..1717
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.
Gus Wiseman, Non-isomorphic representatives of the a(1) = 2 through a(5) = 35 connected multigraphs with loops with multiset density -1.
Index entries for sequences related to trees

Essentially the same as A036251.

Cf. A000055, A007718, A007719, A038052, A191646, A303837, A321155, A321229, A321254, A321256, A322111.

max = 30; B[] = 1; Do[B[x] = x*Exp[Sum[(B[x^k] + x^k)/k + O[x]^n, {k, 1, n}]] // Normal, {n, 1, max}]; A[x_] = B[x] - B[x]^2/2 + B[x^2]/2; CoefficientList[1 + A[x] + O[x]^max, x] (* Jean-François Alcover, Jan 28 2019 *)

G.f.: B(x) - B^2(x)/2 + B(x^2)/2, where B(x) is g.f. for A036249.

A362389 G.f. satisfies A(x) = exp( Sum_{k>=1} (2^k + A(x^k)) * x^k/k ).

Original entry on oeis.org

1, 3, 10, 34, 122, 450, 1723, 6758, 27135, 110913, 460395, 1935233, 8222504, 35255000, 152353021, 662892684, 2901595559, 12768195617, 56450822365, 250637657015, 1117060889815, 4995815027658, 22413020866875, 100842092305575, 454912716037387

Offset: 0

Seiichi Manyama, Jun 09 2023

nonn

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

Cf. A001678, A036249, A363541.

Cf. A029857, A363542, A363545.

seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, (2^k+subst(A, x, x^k))*x^k/k)+x*O(x^n))); Vec(A);

A(x) = B(x)/(1 - 2*x) where B(x) is the g.f. of A363545.
A(x) = Sum_{k>=0} a(k) * x^k = 1/(1-2*x) * 1/Product_{k>=0} (1-x^(k+1))^a(k).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( 2^k + Sum_{d|k} d * a(d-1) ) * a(n-k).

A363541 G.f. satisfies A(x) = exp( Sum_{k>=1} (3^k + A(x^k)) * x^k/k ).

Original entry on oeis.org

1, 4, 17, 73, 324, 1469, 6838, 32490, 157398, 775010, 3870690, 19567202, 99957231, 515250057, 2676884745, 14002926871, 73693381322, 389904743248, 2072794614996, 11066421965311, 59310040841395, 318978744562253, 1720962766007827

Offset: 0

Seiichi Manyama, Jun 09 2023

nonn

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

Cf. A001678, A036249, A362389.

Cf. A363507, A363543, A363546.

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

A(x) = B(x)/(1 - 3*x) where B(x) is the g.f. of A363546.
A(x) = Sum_{k>=0} a(k) * x^k = 1/(1-3*x) * 1/Product_{k>=0} (1-x^(k+1))^a(k).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( 3^k + Sum_{d|k} d * a(d-1) ) * a(n-k).

A363507 G.f. satisfies A(x) = exp( Sum_{k>=1} (3 + A(x^k)) * x^k/k ).

Original entry on oeis.org

1, 4, 14, 50, 191, 763, 3180, 13640, 59937, 268304, 1219626, 5614038, 26117296, 122598622, 579977691, 2762264225, 13234003724, 63737225733, 308406648979, 1498558628584, 7309116199687, 35772044402485, 175621484712091, 864670723348447

Offset: 0

Seiichi Manyama, Jun 06 2023

nonn

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

Cf. A000081, A029857, A036249, A363508.

Cf. A363509.

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

A(x) = Sum_{k>=0} a(k) * x^k = 1/(1-x)^3 * 1/Product_{k>=0} (1-x^(k+1))^a(k).

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( 3 + Sum_{d|k} d * a(d-1) ) * a(n-k).

A038051 G.f.: B(x/(1-x)) where B is g.f. of A000169.

Original entry on oeis.org

1, 3, 14, 98, 944, 11642, 175108, 3108310, 63601168, 1473864722, 38152990484, 1091172974102, 34169139856024, 1162736848398010, 42723615842296540, 1685853467536076798, 71101435046807892512, 3191843270961299033762, 151956292916451992949028

Offset: 1

Christian G. Bower, Jan 04 1999

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A036249, A038052.

E.g.f. of A048802.

CoefficientList[Series[E^x*(-LambertW[-x]/(1+LambertW[-x])/x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Feb 17 2014 *)

E.g.f.: int(exp(x)*(-LambertW(-x)/(1+LambertW(-x))/x), x). a(n) = Sum_{k=0..n-1} binomial(n-1, k)*(k+1)^k. - Vladeta Jovovic, Apr 12 2003

a(n) ~ n^(n-1) * exp(exp(-1)). - Vaclav Kotesovec, Feb 17 2014

Corrected by Christian G. Bower, Mar 15 1999

A303911 Triangle T(w>=1,1<=n<=w) read by rows: the number of rooted weighted trees with n nodes and weight w.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 4, 1, 4, 10, 13, 9, 1, 5, 16, 31, 35, 20, 1, 6, 24, 60, 98, 95, 48, 1, 7, 33, 103, 217, 304, 262, 115, 1, 8, 44, 162, 423, 764, 945, 727, 286, 1, 9, 56, 241, 743, 1658, 2643, 2916, 2033, 719, 1, 10, 70, 341, 1221, 3224, 6319, 8996, 8984, 5714, 1842, 1, 11, 85, 466, 1893

Offset: 1

R. J. Mathar, May 02 2018

Weights are positive integer labels on the nodes. The weight of the tree is the sum of the weights of its nodes.

			The triangle starts
1 ;
1  1 ;
1  2  2 ;
1  3  5   4 ;
1  4 10  13    9 ;
1  5 16  31   35    20 ;
1  6 24  60   98    95    48 ;
1  7 33 103  217   304   262   115 ;
The first column (for a single node n=1) is 1, because all the weight is on that node.

Andrew Howroyd, Table of n, a(n) for n = 1..1275
F. Harary, G. Prins, The number of homeomorphically irreducible trees and other species, Acta Math. 101 (1959) 141-162, W(x,y) equation (9a)

Cf. A000081 (diagonal), A000107 (subdiagonal), A036249 (row sums), A303841 (not rooted).

EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
seq(n)={my(v=[1]); for(i=2, n, v=concat([1], v + EulerMT(y*v))); v}
{my(A=seq(10)); for(n=1, #A, print(Vecrev(A[n])))} \\ Andrew Howroyd, May 19 2018

A363508 G.f. satisfies A(x) = exp( Sum_{k>=1} (4 + A(x^k)) * x^k/k ).

Original entry on oeis.org

1, 5, 20, 80, 340, 1516, 7046, 33736, 165436, 826566, 4193348, 21542664, 111848161, 585949358, 3093526496, 16442687695, 87914559018, 472522551440, 2551591234444, 13836226412386, 75311992329508, 411336641019998, 2253641429297336

Offset: 0

Seiichi Manyama, Jun 06 2023

nonn

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

Cf. A000081, A029857, A036249, A363507.

Cf. A363510.

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

A(x) = Sum_{k>=0} a(k) * x^k = 1/(1-x)^4 * 1/Product_{k>=0} (1-x^(k+1))^a(k).

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( 4 + Sum_{d|k} d * a(d-1) ) * a(n-k).

cp's OEIS Frontend

A048802 Number of labeled rooted trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

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A052855 Number of forests of rooted trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

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A029856 Number of rooted trees with 2-colored leaves.

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A036250 Number of trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

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A362389 G.f. satisfies A(x) = exp( Sum_{k>=1} (2^k + A(x^k)) * x^k/k ).

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A363541 G.f. satisfies A(x) = exp( Sum_{k>=1} (3^k + A(x^k)) * x^k/k ).

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A363507 G.f. satisfies A(x) = exp( Sum_{k>=1} (3 + A(x^k)) * x^k/k ).

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A038051 G.f.: B(x/(1-x)) where B is g.f. of A000169.

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