Displaying 1-10 of 19 results found.
Triangle read by rows: T(n,k) is the number of inequivalent colorings of lone-child-avoiding rooted trees with n colored leaves using exactly k colors.
+10
45
1, 1, 1, 2, 3, 2, 5, 17, 12, 5, 12, 73, 95, 44, 12, 33, 369, 721, 512, 168, 33, 90, 1795, 5487, 5480, 2556, 625, 90, 261, 9192, 41945, 58990, 36711, 12306, 2342, 261, 766, 47324, 321951, 625088, 516952, 224241, 57155, 8702, 766, 2312, 249164, 2483192, 6593103, 7141755, 3965673, 1283624, 258887, 32313, 2312
COMMENTS
Only the leaves are colored. Equivalence is up to permutation of the colors.
Lone-child-avoiding rooted trees are also called planted series-reduced trees in some other sequences.
EXAMPLE
Triangle begins:
1;
1, 1;
2, 3, 2;
5, 17, 12, 5;
12, 73, 95, 44, 12;
33, 369, 721, 512, 168, 33;
90, 1795, 5487, 5480, 2556, 625, 90;
261, 9192, 41945, 58990, 36711, 12306, 2342, 261;
766, 47324, 321951, 625088, 516952, 224241, 57155, 8702, 766;
...
Non-isomorphic representatives of the 39 = 5 + 17 + 12 + 5 trees with four colored leaves:
(1111) (1112) (1123) (1234)
(1(111)) (1122) (1(123)) (1(234))
(11(11)) (1(112)) (11(23)) (12(34))
((11)(11)) (11(12)) (12(13)) ((12)(34))
(1(1(11))) (1(122)) (2(113)) (1(2(34)))
(11(22)) (23(11))
(12(11)) ((11)(23))
(12(12)) (1(1(23)))
(2(111)) ((12)(13))
((11)(12)) (1(2(13)))
(1(1(12))) (2(1(13)))
((11)(22)) (2(3(11)))
(1(1(22)))
(1(2(11)))
((12)(12))
(1(2(12)))
(2(1(11)))
(End)
PROG
(PARI) \\ See link above for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sExp(x*Ser(v[1..n])), n )); x*Ser(v)}
{my(A=InequivalentColoringsTriangle(cycleIndexSeries(10))); for(n=1, #A~, print(A[n, 1..n]))}
CROSSREFS
The case with only one color is A000669. A000311 counts singleton-reduced phylogenetic trees.
A001678 counts unlabeled lone-child-avoiding rooted trees.
A005804 counts phylogenetic rooted trees with n labels.
A060356 counts labeled lone-child-avoiding rooted trees.
A141268 counts lone-child-avoiding rooted trees with leaves summing to n.
A291636 lists Matula-Goebel numbers of lone-child-avoiding rooted trees.
A316651 counts lone-child-avoiding rooted trees with normal leaves.
A316652 counts lone-child-avoiding rooted trees with strongly normal leaves.
A330465 counts inequivalent leaf-colorings of phylogenetic rooted trees.
Expansion of e.g.f.: -LambertW(-x/(1+x)).
+10
33
0, 1, 0, 3, 4, 65, 306, 4207, 38424, 573057, 7753510, 134046671, 2353898196, 47602871329, 1013794852266, 23751106404495, 590663769125296, 15806094859299329, 448284980183376078, 13515502344669830287
COMMENTS
Also the number of labeled lone-child-avoiding rooted trees with n nodes. A rooted tree is lone-child-avoiding if it has no unary branchings, meaning every non-leaf node covers at least two other nodes. The unlabeled version is A001678(n + 1). - Gus Wiseman, Jan 20 2020
FORMULA
a(n) = Sum_{k=1..n} (-1)^(n-k)*n!/k!*binomial(n-1, k-1)*k^(k-1). a(n) = Sum_{k=0..n} Stirling1(n, k)* A058863(k). - Vladeta Jovovic, Sep 17 2003
EXAMPLE
Non-isomorphic representatives of the a(7) = 4207 trees, written as root[branches], are:
1[2,3[4,5[6,7]]]
1[2,3[4,5,6,7]]
1[2[3,4],5[6,7]]
1[2,3,4[5,6,7]]
1[2,3,4,5[6,7]]
1[2,3,4,5,6,7]
(End)
MAPLE
seq(coeff(series( -LambertW(-x/(1+x)), x, n+1), x, n)*n!, n = 0..20); # G. C. Greubel, Mar 16 2020
MATHEMATICA
CoefficientList[Series[-LambertW[-x/(1+x)], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *) sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
a[n_]:=If[n==1, 1, n*Sum[Times@@a/@Length/@stn, {stn, Select[sps[Range[n-1]], Length[#]>1&]}]];
PROG
(PARI) { for (n=0, 100, f=n!; a=sum(k=1, n, (-1)^(n - k)*f/k!*binomial(n - 1, k - 1)*k^(k - 1)); write("b060356.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 04 2009 (PARI) my(x='x+O('x^20)); concat([0], Vec(serlaplace(-lambertw(-x/(1+x))))) \\ G. C. Greubel, Feb 19 2018 (GAP) List([0..20], n->Sum([1..n], k->(-1)^(n-k)*Factorial(n)/Factorial(k) *Binomial(n-1, k-1)*k^(k-1))); # Muniru A Asiru, Feb 19 2018
CROSSREFS
The unlabeled version is A001678(n + 1). The case where the root is fixed is A108919. Unlabeled rooted trees are counted by A000081. Lone-child-avoiding rooted trees with labeled leaves are A000311. Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636. Singleton-reduced rooted trees are counted by A330951. Cf. A000669, A004111, A005121, A048816, A292504, A316651, A316652, A318231, A318813, A330465, A330624.
Number of series-reduced planted trees with n leaves of 2 colors.
+10
19
2, 3, 10, 40, 170, 785, 3770, 18805, 96180, 502381, 2667034, 14351775, 78096654, 429025553, 2376075922, 13252492311, 74372374366, 419651663108, 2379399524742, 13549601275893, 77460249369658, 444389519874841
COMMENTS
Consider the free algebraic system with two commutative associative operators (x+y) and (x*y) and two generators A,B. The number of elements with n occurrences of the generators is 2*a(n) if n>1, and the number of generators if n=1. - Michael Somos, Aug 07 2017 Also the number of semi-lone-child-avoiding rooted trees with n leaves. Semi-lone-child-avoiding means there are no vertices with exactly one child unless that child is an endpoint/leaf. For example, the a(1) = 2 through a(3) = 10 trees are:
o (oo) (ooo)
(o) (o(o)) (o(oo))
((o)(o)) (oo(o))
((o)(oo))
(o(o)(o))
(o(o(o)))
((o)(o)(o))
((o)(o(o)))
(o((o)(o)))
((o)((o)(o)))
(End)
FORMULA
Doubles (index 2+) under EULER transform.
Product_{k>=1} (1-x^k)^-a(k) = 1 + a(1)*x + Sum_{k>=2} 2*a(k)*x^k. - Michael Somos, Aug 07 2017 a(n) ~ c * d^n / n^(3/2), where d = 6.158893517087396289837838459951206775682824030495453326610366016992093939... and c = 0.1914250508201011360729769525164141605187995730026600722369002... - Vaclav Kotesovec, Aug 17 2018
EXAMPLE
For n=2, the 2*a(2) = 6 elements are: A+A, A+B, B+B, A*A, A*B, B*B. - Michael Somos, Aug 07 2017
MATHEMATICA
terms = 22;
B[x_] = x O[x]^(terms+1);
A[x_] = 1/(1 - x + B[x])^2;
Do[A[x_] = A[x]/(1 - x^k + B[x])^Coefficient[A[x], x, k] + O[x]^(terms+1) // Normal, {k, 2, terms+1}];
slaurte[n_]:=If[n==1, {o, {o}}, Join@@Table[Union[Sort/@Tuples[slaurte/@ptn]], {ptn, Rest[IntegerPartitions[n]]}]];
Table[Length[slaurte[n]], {n, 10}] (* Gus Wiseman, Feb 07 2020 *)
PROG
(PARI) {a(n) = my(A, B); if( n<2, 2*(n>0), B = x * O(x^n); A = 1 / (1 - x + B)^2; for(k=2, n, A /= (1 - x^k + B)^polcoeff(A, k)); polcoeff(A, n)/2)}; /* Michael Somos, Aug 07 2017 */
CROSSREFS
Lone-child-avoiding rooted trees with n leaves are A000669. Lone-child-avoiding rooted trees with n vertices are A001678. The locally disjoint case is A331874. Semi-lone-child-avoiding rooted trees with n vertices are A331934. Matula-Goebel numbers of these trees are A331935.
Number of semi-lone-child-avoiding rooted trees with n unlabeled vertices.
+10
19
1, 1, 1, 2, 4, 7, 15, 29, 62, 129, 279, 602, 1326, 2928, 6544, 14692, 33233, 75512, 172506, 395633, 911108, 2105261, 4880535, 11346694, 26451357, 61813588, 144781303, 339820852, 799168292, 1882845298, 4443543279, 10503486112, 24864797324, 58944602767, 139918663784
COMMENTS
A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf.
FORMULA
Product_{k > 0} 1/(1 - x^k)^a(k) = A(x) + A(x)/x - x where A(x) = Sum_{k > 0} x^k a(k).
Euler transform is b(1) = 1, b(n > 1) = a(n) + a(n + 1).
EXAMPLE
The a(1) = 1 through a(7) = 15 trees:
o (o) (oo) (ooo) (oooo) (ooooo) (oooooo)
(o(o)) (o(oo)) (o(ooo)) (o(oooo))
(oo(o)) (oo(oo)) (oo(ooo))
((o)(o)) (ooo(o)) (ooo(oo))
((o)(oo)) (oooo(o))
(o(o)(o)) ((o)(ooo))
(o(o(o))) ((oo)(oo))
(o(o)(oo))
(o(o(oo)))
(o(oo(o)))
(oo(o)(o))
(oo(o(o)))
((o)(o)(o))
((o)(o(o)))
(o((o)(o)))
MATHEMATICA
sse[n_]:=Switch[n, 1, {{}}, 2, {{{}}}, _, Join@@Function[c, Union[Sort/@Tuples[sse/@c]]]/@Rest[IntegerPartitions[n-1]]];
Table[Length[sse[n]], {n, 10}]
PROG
(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=[1, 1]); for(n=2, n-1, v=concat(v, EulerT(v)[n] - v[n])); v} \\ Andrew Howroyd, Feb 09 2020
CROSSREFS
The same trees counted by leaves are A050381. The locally disjoint version is A331872. Matula-Goebel numbers of these trees are A331935. Lone-child-avoiding rooted trees are A001678. Cf. A000081, A000669, A198518, A289501, A291636, A306200, A320268, A330465, A330951, A331873, A331874, A331933, A331966.
Number of non-isomorphic series/singleton-reduced rooted trees on a multiset of size n.
+10
12
1, 1, 2, 7, 39, 236, 1836, 16123, 162008, 1802945, 22012335, 291290460, 4144907830, 62986968311, 1016584428612, 17344929138791, 311618472138440, 5875109147135658, 115894178676866576, 2385755803919949337, 51133201045333895149, 1138659323863266945177, 26296042933904490636133
COMMENTS
A series/singleton-reduced rooted tree on a multiset m is either the multiset m itself or a sequence of series/singleton-reduced rooted trees, one on each part of a multiset partition of m that is neither minimal (all singletons) nor maximal (only one part).
EXAMPLE
Non-isomorphic representatives of the a(4) = 39 trees, with singleton leaves (x) replaced by just x:
(1111) (1112) (1122) (1123) (1234)
(1(111)) (1(112)) (1(122)) (1(123)) (1(234))
(11(11)) (11(12)) (11(22)) (11(23)) (12(34))
((11)(11)) (12(11)) (12(12)) (12(13)) ((12)(34))
(1(1(11))) (2(111)) ((11)(22)) (2(113)) (1(2(34)))
((11)(12)) (1(1(22))) (23(11))
(1(1(12))) ((12)(12)) ((11)(23))
(1(2(11))) (1(2(12))) (1(1(23)))
(2(1(11))) ((12)(13))
(1(2(13)))
(2(1(13)))
(2(3(11)))
PROG
(PARI) \\ See links in A339645 for combinatorial species functions. cycleIndexSeries(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sEulerT(x*Ser(v[1..n])), n )); x*Ser(v)}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 11 2020
CROSSREFS
The case with all atoms equal or all atoms different is A000669. Not requiring singleton-reduction gives A330465. Labeled versions are A316651 (normal orderless) and A330471 (strongly normal). The case where the leaves are sets is A330626. Cf. A000311, A005121, A005804, A141268, A213427, A292504, A292505, A318812, A318848, A318849, A330467, A330469, A330474, A330624.
Number of lone-child-avoiding locally disjoint rooted trees whose leaves are positive integers summing to n, with no two distinct leaves directly under the same vertex.
+10
12
1, 2, 3, 8, 16, 48, 116, 341, 928, 2753, 7996, 24254, 73325, 226471, 702122
COMMENTS
A tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other unequal child of the same vertex. It is lone-child-avoiding if there are no unary branchings.
EXAMPLE
The a(1) = 1 through a(5) = 16 trees:
1 2 3 4 5
(11) (111) (22) (11111)
(1(11)) (1111) ((11)3)
(2(11)) (1(22))
(1(111)) (2(111))
(11(11)) (1(1111))
((11)(11)) (11(111))
(1(1(11))) (111(11))
(1(2(11)))
(2(1(11)))
(1(1(111)))
(1(11)(11))
(1(11(11)))
(11(1(11)))
(1((11)(11)))
(1(1(1(11))))
MATHEMATICA
disjointQ[u_]:=Apply[And, Outer[#1==#2||Intersection[#1, #2]=={}&, u, u, 1], {0, 1}];
usot[n_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[usot/@ptn]], disjointQ[DeleteCases[#, _?AtomQ]]&&SameQ@@Select[#, AtomQ]&], {ptn, Select[IntegerPartitions[n], Length[#]>1&]}], n];
Table[Length[usot[n]], {n, 12}]
CROSSREFS
The non-locally disjoint version is A141268. Locally disjoint trees counted by vertices are A316473. The case where all leaves are 1's is A316697. Number of trees counted by A331678 with all atoms equal to 1. Matula-Goebel numbers of locally disjoint rooted trees are A316495. Unlabeled lone-child-avoiding locally disjoint rooted trees are A331680. Cf. A000081, A000669, A001678, A005804, A060356, A300660, A316471, A316694, A316696, A319312, A330465, A331681.
Number of lone-child-avoiding locally disjoint unlabeled rooted trees with n vertices.
+10
12
1, 0, 1, 1, 2, 3, 6, 9, 16, 26, 45, 72, 124, 201, 341, 561, 947, 1571, 2651, 4434, 7496, 12631, 21423, 36332, 61910, 105641, 180924, 310548, 534713, 923047
COMMENTS
A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other unequal child of the same vertex. Lone-child-avoiding means there are no unary branchings.
EXAMPLE
The a(1) = 1 through a(9) = 16 trees (empty column indicated by dot):
o . (oo) (ooo) (oooo) (ooooo) (oooooo) (ooooooo) (oooooooo)
(o(oo)) (o(ooo)) (o(oooo)) (o(ooooo)) (o(oooooo))
(oo(oo)) (oo(ooo)) (oo(oooo)) (oo(ooooo))
(ooo(oo)) (ooo(ooo)) (ooo(oooo))
((oo)(oo)) (oooo(oo)) (oooo(ooo))
(o(o(oo))) (o(o(ooo))) (ooooo(oo))
(o(oo)(oo)) ((ooo)(ooo))
(o(oo(oo))) (o(o(oooo)))
(oo(o(oo))) (o(oo(ooo)))
(o(ooo(oo)))
(oo(o(ooo)))
(oo(oo)(oo))
(oo(oo(oo)))
(ooo(o(oo)))
(o((oo)(oo)))
(o(o(o(oo))))
MATHEMATICA
disjointQ[u_]:=Apply[And, Outer[#1==#2||Intersection[#1, #2]=={}&, u, u, 1], {0, 1}];
strut[n_]:=If[n==1, {{}}, Select[Join@@Function[c, Union[Sort/@Tuples[strut/@c]]]/@Rest[IntegerPartitions[n-1]], disjointQ]];
Table[Length[strut[n]], {n, 10}]
CROSSREFS
The Matula-Goebel numbers of these trees are A331871. The non-locally disjoint version is A001678. These trees counted by number of leaves are A316697. The semi-lone-child-avoiding version is A331872. Cf. A000081, A000669, A005804, A060356, A141268, A300660, A316471, A316473, A316694, A316495, A319312, A330465, A331679, A331681, A331683.
Number of series-reduced labeled trees with n nodes.
+10
11
1, 0, 1, 1, 13, 51, 601, 4803, 63673, 775351, 12186061, 196158183, 3661759333, 72413918019, 1583407093633, 36916485570331, 929770285841137, 24904721121298671, 711342228666833173, 21502519995056598639, 687345492498807434461, 23135454269839313430715, 818568166383797223246601, 30357965273255025673685091
COMMENTS
"Series-reduced" means that if the tree is rooted at 1, then there is no node with just a single child.
Callan points out that A002792 is an incorrect version of this sequence. - Joerg Arndt, Jul 01 2014
FORMULA
1 = Sum_{n>=0} a(n+1)*(exp(x)-x)^(-n-1)*x^n/n!.
E.g.f.: A(x) = Sum_{n>=0} a(n+1)*x^n/n! satisfies A(x) = exp(x*A(x))/(1+x). - Olivier Gérard, Dec 31 2013 (edited by Gus Wiseman, Dec 31 2019) E.g.f.: -Integral (LambertW(-x/(1 + x))/x) dx. - Ilya Gutkovskiy, Jul 01 2020
MATHEMATICA
f[n_] := Sum[(-1)^(n-k)*n!/k!*Binomial[n-1, k-1]*k^(k-1), {k, n}]/n; Table[ f[n], {n, 20}] (* Robert G. Wilson v, Jul 21 2005 *)
PROG
(PARI) a(n) = { 1/n * sum(k=1, n, (-1)^(n-k) * binomial(n, k) * (n-1)!/(k-1)! * k^(k-1) ); } \\ Joerg Arndt, Aug 28 2014
CROSSREFS
Cf. A000311, A000669, A001678, A292504, A316651, A316652, A318231, A318813, A330465, A330624, A352410.
Number of series-reduced rooted trees whose leaves are multisets whose multiset union is a strongly normal multiset of size n.
+10
10
1, 1, 4, 18, 154, 1614, 23733, 396190, 8066984, 183930948, 4811382339, 138718632336, 4451963556127, 155416836338920, 5920554613563841, 242873491536944706, 10725017764009207613, 505671090907469848248, 25415190929321149684700, 1354279188424092012064226
COMMENTS
A multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.
Also the number of different colorings of phylogenetic trees with n labels using strongly normal multisets of colors. A phylogenetic tree is a series-reduced rooted tree whose leaves are (usually disjoint) sets.
EXAMPLE
The a(3) = 18 trees:
{1,1,1} {1,1,2} {1,2,3}
{{1},{1,1}} {{1},{1,2}} {{1},{2,3}}
{{1},{1},{1}} {{2},{1,1}} {{2},{1,3}}
{{1},{{1},{1}}} {{1},{1},{2}} {{3},{1,2}}
{{1},{{1},{2}}} {{1},{2},{3}}
{{2},{{1},{1}}} {{1},{{2},{3}}}
{{2},{{1},{3}}}
{{3},{{1},{2}}}
MATHEMATICA
strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n];
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
multing[t_, n_]:=Array[(t+#-1)/#&, n, 1, Times];
amemo[m_]:=amemo[m]=1+Sum[Product[multing[amemo[s[[1]]], Length[s]], {s, Split[c]}], {c, Select[mps[m], Length[#]>1&]}];
Table[Sum[amemo[m], {m, strnorm[n]}], {n, 0, 5}]
PROG
(PARI) \\ See links in A339645 for combinatorial species functions. cycleIndexSeries(n)={my(v=vector(n), p=sExp(x*sv(1) + O(x*x^n))); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sExp(x*Ser(v[1..n])), n ) + polcoef(p, n)); 1 + x*Ser(v)}
StronglyNormalLabelingsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 28 2020
CROSSREFS
The singleton-reduced version is A316652. Not requiring weakly decreasing multiplicities gives A330469. The case where the leaves are sets is A330625. Cf. A000311, A000669, A004114, A005121, A005804, A141268, A292504, A292505, A318812, A318849, A319312, A330471, A330475.
Number of series-reduced rooted trees whose leaves are multisets with a total of n elements covering an initial interval of positive integers.
+10
10
1, 1, 4, 24, 250, 3744, 73408, 1768088, 50468854, 1664844040, 62304622320, 2607765903568, 120696071556230, 6120415124163512, 337440974546042416, 20096905939846645064, 1285779618228281270718, 87947859243850506008984, 6404472598196204610148232
COMMENTS
Also the number of different colorings of phylogenetic trees with n labels using a multiset of colors covering an initial interval of positive integers. A phylogenetic tree is a series-reduced rooted tree whose leaves are (usually disjoint) sets.
EXAMPLE
The a(3) = 24 trees:
(123) (122) (112) (111)
((1)(23)) ((1)(22)) ((1)(12)) ((1)(11))
((2)(13)) ((2)(12)) ((2)(11)) ((1)(1)(1))
((3)(12)) ((1)(2)(2)) ((1)(1)(2)) ((1)((1)(1)))
((1)(2)(3)) ((1)((2)(2))) ((1)((1)(2)))
((1)((2)(3))) ((2)((1)(2))) ((2)((1)(1)))
((2)((1)(3)))
((3)((1)(2)))
MATHEMATICA
allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
multing[t_, n_]:=Array[(t+#-1)/#&, n, 1, Times];
amemo[m_]:=amemo[m]=1+Sum[Product[multing[amemo[s[[1]]], Length[s]], {s, Split[c]}], {c, Select[mps[m], Length[#]>1&]}];
Table[Sum[amemo[m], {m, allnorm[n]}], {n, 0, 5}]
PROG
(PARI)
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=[]); for(n=1, n, v=concat(v, EulerT(concat(v, [binomial(n+k-1, k-1)]))[n])); v}
seq(n)={concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k))))} \\ Andrew Howroyd, Dec 29 2019
CROSSREFS
The singleton-reduced version is A316651. The strongly normal case is A330467. The case when leaves are sets is A330764. Cf. A000311, A000669, A004114, A005121, A005804, A141268, A292504, A292505, A316652, A318812, A318849, A319312, A330625.
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