A000311 -id:A000311 - cp's OEIS frontend

A320294 Number of series-reduced rooted trees whose leaves are non-singleton integer partitions whose multiset union is an integer partition of n with no 1's.

0, 0, 0, 1, 1, 3, 3, 7, 8, 15, 19, 37, 48, 87, 126, 227, 342, 611, 964, 1719, 2806, 4975, 8327, 14782, 25157, 44609, 76972, 136622, 237987, 422881, 742149, 1320825, 2331491, 4156392, 7370868, 13164429, 23433637, 41928557, 74871434, 134203411, 240284935, 431437069

Offset: 1

Gus Wiseman, Oct 09 2018

nonn

Also phylogenetic trees with no singleton leaves on integer partitions of n with no 1's.

			The a(4) = 1 through a(10) = 15 trees:
  (22)  (32)  (33)   (43)   (44)        (54)        (55)
              (42)   (52)   (53)        (63)        (64)
              (222)  (322)  (62)        (72)        (73)
                            (332)       (333)       (82)
                            (422)       (432)       (433)
                            (2222)      (522)       (442)
                            ((22)(22))  (3222)      (532)
                                        ((22)(23))  (622)
                                                    (3322)
                                                    (4222)
                                                    (22222)
                                                    ((22)(24))
                                                    ((22)(33))
                                                    ((23)(23))
                                                    ((22)(222))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000045, A000311, A000669, A002865, A141268, A292504, A304966, A304967, A319312, A320289, A320293, A320295, A320296.

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]]]];
pgtm[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[pgtm/@p]],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Sum[Length[Select[pgtm[m],FreeQ[#,{_}]&]],{m,Select[IntegerPartitions[n],FreeQ[#,1]&]}],{n,10}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=1/prod(k=2, n, 1 - x^k + O(x*x^n)), v=vector(n)); for(n=2, n, v[n]=polcoef(p, n) - 1 + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(16) and beyond from Andrew Howroyd, Oct 25 2018

A320295 Number of series-reduced rooted trees whose leaves are non-singleton integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

0, 1, 2, 5, 8, 19, 34, 80, 165, 394, 892, 2192, 5232, 13057, 32271, 81568, 205748, 525735, 1344828, 3467415, 8960849, 23280323, 60639680, 158559047, 415631368, 1092734050, 2879420753, 7605713020, 20130266302, 53386744298, 141836904569, 377479973474, 1006189769886

Offset: 1

Gus Wiseman, Oct 09 2018

nonn

Also phylogenetic trees with no singleton leaves on integer partitions of n.

			The a(2) = 1 through a(6) = 19 trees:
  (11)  (21)   (22)        (32)         (33)
        (111)  (31)        (41)         (42)
               (211)       (221)        (51)
               (1111)      (311)        (222)
               ((11)(11))  (2111)       (321)
                           (11111)      (411)
                           ((11)(12))   (2211)
                           ((11)(111))  (3111)
                                        (21111)
                                        (111111)
                                        ((11)(13))
                                        ((11)(22))
                                        ((12)(12))
                                        ((11)(112))
                                        ((12)(111))
                                        ((11)(1111))
                                        ((111)(111))
                                        ((11)(11)(11))
                                        ((11)((11)(11)))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000311, A000669, A005804, A141268, A302545, A304966, A319312, A320289, A320294.

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]]]];
pgtm[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[pgtm/@p]],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Sum[Length[Select[pgtm[m],FreeQ[#,{_}]&]],{m,IntegerPartitions[n]}],{n,14}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=1/prod(k=1, n, 1 - x^k + O(x*x^n)), v=vector(n)); for(n=1, n, v[n]=polcoef(p, n) - 1 + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(12) and beyond from Andrew Howroyd, Oct 25 2018

A320296 Number of series-reduced rooted trees whose leaves form an integer partition of n with no 1's.

Original entry on oeis.org

0, 1, 1, 2, 2, 5, 6, 15, 22, 51, 86, 195, 354, 781, 1512, 3286, 6602, 14269, 29424, 63494, 133298, 287909, 612188, 1325375, 2844448, 6176145, 13348858, 29074164, 63187176, 138044144, 301350424, 660265471, 1446678326, 3178246273, 6985464590, 15384556290

Offset: 1

Gus Wiseman, Oct 09 2018

nonn

Also phylogenetic trees with n unlabeled objects and no singleton leaves.

			The a(2) = 1 through a(9) = 22 trees:
2   3   4     5     6        7        8           9
        (22)  (23)  (24)     (25)     (26)        (27)
                    (33)     (34)     (35)        (36)
                    (222)    (223)    (44)        (45)
                    (2(22))  ((22)3)  (224)       (225)
                             (2(23))  (233)       (234)
                                      (2222)      (333)
                                      ((22)4)     (2223)
                                      (2(24))     ((22)5)
                                      ((23)3)     (2(25))
                                      (2(33))     ((23)4)
                                      (2(222))    (2(34))
                                      (22(22))    ((24)3)
                                      ((22)(22))  ((33)3)
                                      (2(2(22)))  (2(22)3)
                                                  (2(223))
                                                  (22(23))
                                                  (3(222))
                                                  ((2(22))3)
                                                  ((22)(23))
                                                  (2((22)3))
                                                  (2(2(23)))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000311, A000669, A002865, A005804, A141268, A304967, A319312, A320293.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
t[n_]:=t[n]=If[PrimeQ[n],{n},Join@@Table[Union[Sort/@Tuples[t/@fac]],{fac,Select[facs[n],Length[#]>1&]}]];
Table[Sum[Length[t[Times@@Prime/@ptn]],{ptn,Select[IntegerPartitions[n],FreeQ[#,1]&]}],{n,15}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=vector(n)); for(n=2, n, v[n]=1 + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(26) and beyond from Andrew Howroyd, Oct 25 2018

A330654 Number of series/singleton-reduced rooted trees on normal multisets of size n.

Original entry on oeis.org

1, 1, 2, 12, 112, 1444, 24099, 492434, 11913985

Offset: 0

Gus Wiseman, Dec 26 2019

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).
A finite multiset is normal if it covers an initial interval of positive integers.
First differs from A316651 at a(6) = 24099, A316651(6) = 24086. For example, ((1(12))(2(11))) and ((2(11))(1(12))) are considered identical for A316651 (series-reduced rooted trees), but {{{1},{1,2}},{{2},{1,1}}} and {{{2},{1,1}},{{1},{1,2}}} are different series/singleton-reduced rooted trees.

			The a(0) = 1 through a(3) = 12 trees:
  {}  {1}  {1,1}  {1,1,1}
           {1,2}  {1,1,2}
                  {1,2,2}
                  {1,2,3}
                  {{1},{1,1}}
                  {{1},{1,2}}
                  {{1},{2,2}}
                  {{1},{2,3}}
                  {{2},{1,1}}
                  {{2},{1,2}}
                  {{2},{1,3}}
                  {{3},{1,2}}

The orderless version is A316651.
The strongly normal case is A330471.
The unlabeled version is A330470.
The balanced version is A330655.
The case with all atoms distinct is A000311.
The case with all atoms equal is A196545.
Normal multiset partitions are A255906.
Cf. A000669, A004114, A005804, A281118, A316651, A330469, A330626, A330676.

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]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
ssrtrees[m_]:=Prepend[Join@@Table[Tuples[ssrtrees/@p],{p,Select[mps[m],Length[m]>Length[#1]>1&]}],m];
Table[Sum[Length[ssrtrees[s]],{s,allnorm[n]}],{n,0,5}]

A007151 Number of planted evolutionary trees of magnitude n.

Original entry on oeis.org

1, 3, 19, 198, 2906, 55018, 1275030, 34947664, 1105740320, 39661089864, 1590232358584, 70482038536880, 3421732373367504, 180574681050278960, 10292371442183694832, 630125771602386523392, 41239934114630205030656

Offset: 1

Robert W. Robinson

Also number of labeled rooted trees with n generators. (A generator is a leaf or a node with just one child.) - Christian G. Bower, Jun 07 2005

L. R. Foulds and R. W. Robinson, Counting certain classes of evolutionary trees with singleton labels, Congress. Num., 44 (1984), 65-88.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..360
L. R. Foulds and R. W. Robinson, Counting certain classes of evolutionary trees with singleton labels, Congress. Num., 44 (1984), 65-88. (Annotated scanned copy)
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A007152, A108521, A108522, A000169, A000311.

A007151 := proc(n)
    local k,j,i,m ,a;
    if n =1 then
        1;
    else
        a := 0 ;
        for k from 1 to n-1 do
        for j from 1 to k do
        for i from 0 to n-1 do
        for m from 0 to j do
             a := a+(n+k-1)! /(k-j)! *binomial(j+i-1,j-1) *2^m *(-1)^(m+i) *combinat[stirling2](n-m+j-i-1,j-m) / m! /(n-m+j-i-1)! ;
        end do:
        end do:
        end do:
        end do:
        a ;
    end if;
end proc:
seq(A007151(n),n=1..10) ; # R. J. Mathar, Mar 19 2018

Rest[CoefficientList[InverseSeries[Series[(1 - E^x + 2*x)/(1 + x),{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 08 2014 *)

a(n):=if n=1 then 1 else (sum((n+k-1)!*sum(1/((k-j)!)*sum(binomial(j+i-1,j-1)*sum((2^m*(-1)^(m+i)*stirling2(n-m+j-i-1,j-m))/(m!*(n-m+j-i-1)!),m,0,j),i,0,n-1),j,1,k),k,1,n-1)); /* Vladimir Kruchinin, Aug 07 2012 */

for(n=1,20, print1(if(n==1,1,sum(k=1,n-1, (n+k-1)!*sum(j=1,k, (1/(k-j)!)* sum(i=0,n-1, binomial(j+i-1,j-1)*sum(m=0,j, 2^m*(-1)^(m+i)* stirling(n-m+j-i-1,j-m,2)/(m!*(n-m+j-i-1)!)))))), ", ")) \\ G. C. Greubel, Nov 26 2017

E.g.f. satisfies (2-x)*A(x) = x - 1 + exp(A(x)). - Christian G. Bower, Jun 07 2005
a(n) = Sum_{k=1..(n-1)} (n+k-1)!*Sum_{j=1..k} (1/(k-j)!)*Sum_{i=0..(n-1)} binomial(j+i-1,j-1)*Sum_{m=0..j} 2^m*(-1)^(m+i)*Stirling2(n-m+j-i-1,j-m)/(m!*(n-m+j-i-1)!), n>1, a(1)=1. - Vladimir Kruchinin, Aug 07 2012
a(n) ~ sqrt(LambertW(1)+1) * n^(n-1) * (LambertW(1))^n / (exp(n) * (2*LambertW(1)-1)^(n-1/2)). - Vaclav Kotesovec, Jan 08 2014

A052526 Number of labeled rooted trees with n leaves in which the degrees of the root and all internal nodes are >= 3.

Original entry on oeis.org

0, 0, 0, 1, 1, 11, 36, 372, 2311, 26252, 243893, 3173281, 38916879, 583922418, 8808814262, 151530476047, 2694658394356, 52607648010035, 1072975736368359, 23516009286474813, 539838208864165036

Offset: 0

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

Old name was "Non-planar labeled trees with neither unary nor binary nodes". "Non-planar" presumably indicates that we are only concerned with the abstract tree, not with a particular embedding in the plane.

			For n=5 there are 2 unlabeled trees of this type. In the first, the root node has 5 children which are all leaves. In the second, the root has 3 children; 2 are leaves and 1 has 3 children which are leaves. The first has only one labeling; the second has binomial(5,2)=10 labelings. So a(5) = 1 + 10 = 11.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 96

Unlabeled trees of this type are counted by A052525. Labeled trees in which the degrees of non-leaf nodes are >= 2 are counted by A000311.

Cf. A226572.

Non spec := [S,{B=Union(S,Z),S=Set(B,3 <= card)},labeled]: seq(combstruct[count](spec, size=n), n=0..20);

ClearAll[a]; max = 20; Z[x_] := Sum[ a[k]*x^k, {k, 0, max}]; a[0] = 0; a[1] = 1/2; a[2] = 0; se = Normal[ Series[ (2*E^Z[x] - 4*Z[x] + 2*x - 2 - Z[x]^2) - x, {x, 0, max}]]; sol = SolveAlways[se == 0, x] // First; A052526 = Join[{0, 0, 0}, Table[k!*2^k*a[k], {k, 3, max}]] /. sol (* _Jean-François Alcover, Sep 25 2012, from first e.g.f. *)
CoefficientList[InverseSeries[Series[1+2*x+1/2*x^2-E^x, {x, 0, 20}], x]-x,x] * Range[0, 20]! (* Vaclav Kotesovec, May 04 2015 *)

a(n):=sum((n+k-1)!*sum(1/(k-j)!*sum((-1)^(l)*sum((2^(l-2*i)*stirling2(n+j-i-l-1,j-l))/(i!*(l-i)!*(n+j-i-l-1)!),i,0,l),l,0,j),j,1,k),k,1,n-1); /* Vladimir Kruchinin, Sep 25 2012 */

E.g.f.: RootOf(2*exp(Z)-4*Z+2*x-2-Z^2)-x.
E.g.f.: reversed[1+2*x+1/2*x^2-exp(x)]-x, a(n):=sum(k=1..n-1, (n+k-1)!*sum(j=1..k, 1/(k-j)!*sum(l=0..j, (-1)^(l)*sum(i=0..l, (2^(l-2*i)*stirling2(n+j-i-l-1,j-l))/(i!*(l-i)!*(n+j-i-l-1)!))))). [Vladimir Kruchinin, Sep 25 2012]
a(n) ~ n^(n-1) / (sqrt(c-1) * exp(n) * (c^2/2 - c - 1)^(n-1/2)), where c = A226572 = -LambertW(-1, -exp(-2)) = 3.146193220620582585237... . - Vaclav Kotesovec, May 04 2015

Edited by Dean Hickerson, Jun 07 2006

A058387 Number of series-parallel networks with n unlabeled edges, multiple edges not allowed.

Original entry on oeis.org

0, 1, 1, 2, 4, 8, 18, 40, 94, 224, 548, 1356, 3418, 8692, 22352, 57932, 151312, 397628, 1050992, 2791516, 7447972, 19950628, 53635310, 144664640, 391358274, 1061628772, 2887113478, 7869761108, 21497678430, 58841838912, 161356288874

Offset: 0

N. J. A. Sloane, Dec 20 2000

This is a series-parallel network: o-o; all other series-parallel networks are obtained by connecting two series-parallel networks in series or in parallel. See A000084 for examples.

Order is not considered significant in series configurations. - Andrew Howroyd, Dec 22 2020

			From _Andrew Howroyd_, Dec 22 2020: (Start)
In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element (an edge) is denoted by 'o'.
a(1) = 1: (o).
a(2) = 1: (oo).
a(3) = 2: (ooo), (o|oo).
a(4) = 4: (oooo), (o(o|oo)), (o|ooo), (oo|oo).
a(5) = 8: (ooooo), (oo(o|oo)), (o(o|ooo)), (o(oo|oo)), (o|oooo), (o|o(o|oo)),  (oo|ooo), (o|oo|oo).
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..500
Steven R. Finch, Series-parallel networks
Steven R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
John W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the sequence v_n).
Index entries for sequences mentioned in Moon (1987)

A000084 is the case that multiple edges are allowed.
A058381 is the case that edges are labeled.
A339290 is the case that order is significant in series configurations.
Cf. A058385, A058386, A000311, A000669, A006351.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(s=p=vector(n)); p[1]=1; for(n=2, n, s[n]=EulerT(p[1..n])[n]; p[n]=vecsum(EulerT(s[1..n])[n-1..n])-s[n]); concat([0], p+s)} \\ Andrew Howroyd, Dec 22 2020

a(n) = A058385(n) + A058386(n).

A181996 Triangle of Ward numbers T(n,k) (n>=0, k=0 if n=0, otherwise 0 <= k <= n-1) read by rows.

Original entry on oeis.org

1, 1, 3, 1, 15, 10, 1, 105, 105, 25, 1, 945, 1260, 490, 56, 1, 10395, 17325, 9450, 1918, 119, 1, 135135, 270270, 190575, 56980, 6825, 246, 1, 2027025, 4729725, 4099095, 1636635, 302995, 22935, 501, 1, 34459425, 91891800, 94594500, 47507460, 12122110, 1487200, 74316, 1012, 1

Offset: 0

N. J. A. Sloane, Apr 05 2012

It appears that the sum of row(n) is A000311(n+1). - Michel Marcus, Feb 07 2013

Conjecture on row sums was proved in the first paragraph of the formula section of the reverse matrix A134991 in 2008 (e.g.f. evaluated at t=1). - Tom Copeland, Jan 03 2016

			Triangle begins:
      1
      1
      3     1
     15    10    1
    105   105   25    1
    945  1260  490   56   1
  10395 17325 9450 1918 119 1 ...

Charles Jordan, Calculus of Finite Differences, Chelsea 1950, p. 172, Table C_{m, i}.

L. Carlitz, The coefficients in an asymptotic expansion and certain related numbers, Duke Math. J., vol 35 (1968), p. 83-90. See page 85.
José L. Cereceda, Figurate numbers and sums of powers of integers, arXiv:2001.03208 [math.NT], 2020. See Table 8 p. 11.
Lane Clark, Asymptotic normality of the Ward numbers, Discrete Math. 203 (1999), no. 1-3, 41-48. MR1696232 (2000d:11101)
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms
Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
Aleks Žigon Tankosič, Recurrence Relations for Some Integer Sequences Related to Ward Numbers, arXiv:2508.04754 [math.CO], 2025. See p. 2.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 1.

See A134991, which is the mirror image and is the main entry for this triangle, for further information.

Cf. A000311.

A181996 := (n,k) -> add((-1)^(n - k + m)*binomial(2*n - k, n + m)*Stirling2(n + m, m), m = 0..n-k):
seq(seq(A181996(n, k), k = 0..n-1+0^n), n=0..8); # Peter Luschny, Feb 19 2021

T(n,k) = {if (n == 0, return(1)); if (k == 0, return (prod(x=2,n, 2*x-1))); if (k == n, return (0)); return((2*n-1-k)*T(n-1,k) + (n-k)*T(n-1, k-1));} \\ Michel Marcus, Feb 07 2013

T(n, k) = Sum_{m = 0..n-k} (-1)^(n - k + m)*C(2*n - k, n + m)*Stirling2(n + m, m). - Peter Luschny, Feb 19 2021

More terms from Michel Marcus, Feb 07 2013

A225170 Number of non-degenerate fanout-free Boolean functions of n variables having AND rank 1.

Original entry on oeis.org

2, 4, 32, 416, 7552, 176128, 5018624, 168968192, 6563282944, 288909131776, 14212910809088, 772776684683264, 46017323176296448, 2978458881388183552, 208198894960190160896, 15631251601179130462208, 1254492810303112820555776, 107174403941451434687463424

Offset: 1

N. J. A. Sloane, Apr 30 2013

nonn

Apart from initial term, same as A005172, which is the main entry for this sequence.

Vaclav Kotesovec, Table of n, a(n) for n = 1..241
J. P. Hayes, Enumeration of fanout-free Boolean functions, J. ACM, 23 (1976), 700-709.
Index entries for sequences related to Boolean functions

Column 1 of A225171.

Cf. A000311, A005172.

max = 16; s = -ProductLog[-Exp[x-1/2]/2] + O[x]^max; Join[{2}, Drop[CoefficientList[s, x]*Range[0, max-1]!, 2]] (* Jean-François Alcover, Oct 18 2016 *)
a[1] = 2; a[n_] := (Sum[(n + k - 1)!*Sum[(-1)^j/(k - j)!*Sum[(-1)^i*2^(n - i + j - 1)*StirlingS1[n - i + j - 1, j - i]/((n - i + j - 1)!*i!), {i, 0, j}], {j, 1, k}], {k, 1, n - 1}]);
Array[a, 20] (* Jean-François Alcover, Jun 24 2018, after Vladimir Kruchinin *)

seq(n) = Vec(serlaplace(serreverse((1 + 2*x - exp(x + O(x*x^n)))/2 ))) \\ Andrew Howroyd, Mar 28 2025

Hayes (1976, Theorem 3) gives a recurrence.
G.f.: 1/Q(0) + 1, where Q(k)= 1 - 2*x*(k+1) - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 18 2013
a(n) ~ (log(2)-1/2)^(1/2 - n) * n^(n-1) / exp(n). - Vaclav Kotesovec, Oct 19 2016
a(n) = 2^n * A000311(n). - Andrew Howroyd, Mar 28 2025

A318847 Number of tree-partitions of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 12, 8, 28, 20, 32, 38, 112, 76, 116, 58, 352, 236, 1296, 176, 540, 288, 4448, 374, 612, 1144, 1812, 824, 16640, 1316, 59968, 612, 2336, 4528, 3208, 2924, 231168, 18320, 10632, 2168, 856960, 7132, 3334400, 3776, 11684, 74080, 12679424, 4919, 19192

Offset: 1

Gus Wiseman, Sep 04 2018

nonn

This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

A tree-partition of m is either m itself or a sequence of tree-partitions, one of each part of a multiset partition of m with at least two parts.

			The a(6) = 6 tree-partitions of {1,1,2}:
  (112)
  ((1)(12))
  ((2)(11))
  ((1)(1)(2))
  ((1)((1)(2)))
  ((2)((1)(1)))

Cf. A000311, A001055, A196545, A281118, A281119, A305936, A318762, A318812, A318813, A318846, A318848.

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]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
allmsptrees[m_]:=Prepend[Join@@Table[Tuples[allmsptrees/@p],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Length[allmsptrees[nrmptn[n]]],{n,20}]

a(n) = A281118(A181821(n)).
a(prime(n)) = A289501(n).
a(2^n) = A005804(n).

More terms from Jinyuan Wang, Jun 26 2020

cp's OEIS Frontend

A320294 Number of series-reduced rooted trees whose leaves are non-singleton integer partitions whose multiset union is an integer partition of n with no 1's.

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A320295 Number of series-reduced rooted trees whose leaves are non-singleton integer partitions whose multiset union is an integer partition of n.

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A320296 Number of series-reduced rooted trees whose leaves form an integer partition of n with no 1's.

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A330654 Number of series/singleton-reduced rooted trees on normal multisets of size n.

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A007151 Number of planted evolutionary trees of magnitude n.

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A052526 Number of labeled rooted trees with n leaves in which the degrees of the root and all internal nodes are >= 3.

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A058387 Number of series-parallel networks with n unlabeled edges, multiple edges not allowed.

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