A001515 -id:A001515 - cp's OEIS frontend

A368726 Number of non-isomorphic connected multiset partitions of weight n into singletons or pairs.

1, 1, 3, 3, 8, 10, 26, 38, 93, 161, 381, 732, 1721, 3566, 8369, 18316, 43280, 98401, 234959, 549628, 1327726, 3175670, 7763500, 18905703, 46762513, 115613599, 289185492, 724438500, 1831398264, 4641907993, 11853385002, 30365353560

Offset: 0

Gus Wiseman, Jan 06 2024

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 10 multiset partitions:
  {{1}}  {{1,1}}    {{1},{1,1}}    {{1,1},{1,1}}      {{1},{1,1},{1,1}}
         {{1,2}}    {{2},{1,2}}    {{1,2},{1,2}}      {{1},{1,2},{2,2}}
         {{1},{1}}  {{1},{1},{1}}  {{1,2},{2,2}}      {{2},{1,2},{1,2}}
                                   {{1,3},{2,3}}      {{2},{1,2},{2,2}}
                                   {{1},{1},{1,1}}    {{2},{1,3},{2,3}}
                                   {{1},{2},{1,2}}    {{3},{1,3},{2,3}}
                                   {{2},{2},{1,2}}    {{1},{1},{1},{1,1}}
                                   {{1},{1},{1},{1}}  {{1},{2},{2},{1,2}}
                                                      {{2},{2},{2},{1,2}}
                                                      {{1},{1},{1},{1},{1}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

For edges of any size we have A007718.
This is the connected case of A320663.
The case of singletons and strict pairs is A368727, Euler transform A339888.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A007716 counts non-isomorphic multiset partitions, into pairs A007717.
A062740 counts connected loop-graphs, unlabeled A054921.
A320732 counts factorizations into primes or semiprimes, strict A339839.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A001515, A000666, A122848, A283877, A302545, A320462, A321405, A368186, A368598, A368599, A368731.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort/@(#/.x_Integer:>s[[x]])]& /@ sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}],Length[Intersection@@s[[#]]]>0&]}, If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n], Max@@Length/@#<=2&&Length[csm[#]]<=1&]]],{n,0,8}]

Inverse Euler transform of A320663.

A368727 Number of non-isomorphic connected multiset partitions of weight n into singletons or strict pairs.

Original entry on oeis.org

1, 1, 2, 2, 5, 6, 15, 21, 49, 82, 184, 341, 766, 1530, 3428, 7249, 16394, 36009, 82492, 186485, 433096, 1001495, 2358182, 5554644, 13255532, 31718030, 76656602, 185982207, 454889643, 1117496012, 2764222322, 6868902152, 17172601190

Offset: 0

Gus Wiseman, Jan 06 2024

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 15 multiset partitions:
  {1}  {12}    {2}{12}    {12}{12}      {2}{12}{12}      {12}{12}{12}
       {1}{1}  {1}{1}{1}  {13}{23}      {2}{13}{23}      {12}{13}{23}
                          {1}{2}{12}    {3}{13}{23}      {13}{23}{23}
                          {2}{2}{12}    {1}{2}{2}{12}    {13}{24}{34}
                          {1}{1}{1}{1}  {2}{2}{2}{12}    {14}{24}{34}
                                        {1}{1}{1}{1}{1}  {1}{2}{12}{12}
                                                         {1}{2}{13}{23}
                                                         {2}{2}{12}{12}
                                                         {2}{2}{13}{23}
                                                         {2}{3}{13}{23}
                                                         {3}{3}{13}{23}
                                                         {1}{1}{2}{2}{12}
                                                         {1}{2}{2}{2}{12}
                                                         {2}{2}{2}{2}{12}
                                                         {1}{1}{1}{1}{1}{1}

Andrew Howroyd, Table of n, a(n) for n = 0..50

For edges of any size we have A056156, with loops A007718.
This is the connected case of A339888.
Allowing loops {x,x} gives A368726, Euler transform A320663.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A007716 counts non-isomorphic multiset partitions, into pairs A007717.
A062740 counts connected loop-graphs, unlabeled A054921.
A320732 counts factorizations into primes or semiprimes, strict A339839.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A001515, A000666, A122848, A283877, A302545, A320462, A321405, A368598, A368599, A368731.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]], {s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={},s,csm[Sort[Append[Delete[s,List /@ c[[1]]],Union@@s[[c[[1]]]]]]]]];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n],And@@UnsameQ@@@#&&Max@@Length/@#<=2&&Length[csm[#]]<=1&]]],{n,0,8}]

Inverse Euler transform of A339888.

A144575 E,g.f.: exp(1-sqrt(1-2x-3x^2)).

Original entry on oeis.org

1, 1, 5, 25, 217, 2321, 32221, 536425, 10547825, 238451617, 6103391221, 174418724921, 5506024371145, 190282381973425, 7145586497798477, 289733147423281801, 12615792602988127201, 587128127734854278465, 29084008051746449028325, 1527881758843209566647897

Offset: 0

N. J. A. Sloane, Jan 07 2009

nonn

Cf. A001515, A144576.

With[{nn=20},CoefficientList[Series[Exp[1-Sqrt[1-2x-3x^2]],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Mar 26 2013 *)

a(n) ~ 3^(n-1/2)*sqrt(2)*n^(n-1)/exp(n-1). - Vaclav Kotesovec, Jun 02 2013

Conjecture D-finite with recurrence: a(n) +(-2*n+3)*a(n-1) +(-3*n^2+12*n-10)*a(n-2) +3*(-2*n+3)*a(n-3) -9*(n-1)*(n-3)*a(n-4)=0. - R. J. Mathar, Jan 24 2020

A144905 a(0) = 1; thereafter a(n) = A105749(n)/n.

Original entry on oeis.org

1, 2, 7, 74, 1596, 58344, 3240840, 254535840, 26862378480, 3667537480320, 629083000385280, 132437508454137600, 33575888768939193600, 10090248381797704243200, 3546915020658948703564800, 1441883923593020355819571200, 671220876625092844683849216000, 354750674999711346878469083136000

Offset: 0

David Applegate and N. J. A. Sloane, Feb 16 2009

nonn

Cf. A001515, A105749, A144906.

B := proc(n, k, M) local i; option remember;
if n = k then 1;
elif k < n then 0;
elif n < 1 then 0;
else
add( binomial(k-1,i)*B(n-1,k-1-i,M),i=0..M-1 );
end if;
end proc;
p:=proc(n,M) add(B(n,k,M),k=0..M*n); end;
l:=proc(n,M) n!*p(n,M); end;
[seq(l(n,2)/n,n=1..30)];

A144906 a(0) = 1; thereafter a(n) = A144422(n)/n.

Original entry on oeis.org

1, 3, 31, 1684, 271776, 97484904, 65617109160, 74248657560720, 130752443907524880, 338450307621257099520, 1232284889962378714855680, 6094200542431309662145478400, 39788645361978802435089535468800, 334957784448996146804912925763507200

Offset: 0

David Applegate and N. J. A. Sloane, Feb 16 2009

nonn

Cf. A144422, A001515, A105749, A144905.

A185296 Triangle of connection constants between the falling factorials (x)(n) and (2*x)(n).

Original entry on oeis.org

1, 0, 2, 0, 2, 4, 0, 0, 12, 8, 0, 0, 12, 48, 16, 0, 0, 0, 120, 160, 32, 0, 0, 0, 120, 720, 480, 64, 0, 0, 0, 0, 1680, 3360, 1344, 128, 0, 0, 0, 0, 1680, 13440, 13440, 3584, 256, 0, 0, 0, 0, 0, 30240, 80640, 48384, 9216, 512

Offset: 0

Peter Bala, Feb 20 2011

The falling factorial polynomials (x)_n := x*(x-1)*...*(x-n+1), n = 0,1,2,..., form a basis for the space of polynomials. Hence the polynomial (2*x)_n may be expressed as a linear combination of x_0, x_1,...,x_n; the coefficients in the expansion form the n-th row of the table. Some examples are given below.
This triangle is connected to two families of orthogonal polynomials, the Hermite polynomials H(n,x) A060821, and the Bessel polynomials y(n,x)A001498. The first few Hermite polynomials are
... H(0,x) = 1
... H(1,x) = 2*x
... H(2,x) = -2+4*x^2
... H(3,x) = -12*x+8*x^3
... H(4,x) = 12-48*x^2+16*x^4.
The unsigned coefficients of H(n,x) give the nonzero entries of the n-th row of the triangle.
The Bessel polynomials y(n,x) begin
... y(0,x) = 1
... y(1,x) = 1+x
... y(2,x) = 1+3*x+3*x^2
... y(3,x) = 1+6*x+15*x^2+15*x^3.
The entries in the n-th column of this triangle are the coefficients of the scaled Bessel polynomials 2^n*y(n,x).
Also the Bell transform of g(n) = 2 if n<2 else 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			Triangle begins
n\k|...0.....1.....2.....3.....4.....5.....6
============================================
0..|...1
1..|...0.....2
2..|...0.....2.....4
3..|...0.....0....12.....8
4..|...0.....0....12....48....16
5..|...0.....0.....0...120...160....32
6..|...0.....0.....0...120...720...480....64
..
Row 3:
(2*x)_3 = (2*x)*(2*x-1)*(2*x-2) = 8*x*(x-1)*(x-2) + 12*x*(x-1).
Row 4:
(2*x)_4 = (2*x)*(2*x-1)*(2*x-2)*(2*x-3) = 16*x*(x-1)*(x-2)*(x-3) +
48*x*(x-1)*(x-2)+ 12*x*(x-1).
Examples of recurrence relation
T(4,4) = 5*T(3,4) + 2*T(3,3) = 5*0 + 2*8 = 16;
T(5,4) = 4*T(4,4) + 2*T(4,3) = 4*16 + 2*48 = 160;
T(6,4) = 3*T(5,4) + 2*T(5,3) = 3*160 + 2*120 = 720;
T(7,4) = 2*T(6,4) + 2*T(6,3) = 2*720 + 2*120 = 1680.

L. Comtet, Advanced Combinatorics, Reidel, 1974, page 158, exercise 7.

Cf. A000898 (row sums), A001498, A001515, A059343, A060821.

T := (n,k) -> (n!/k!)*binomial(k,n-k)*2^(2*k-n):
seq(seq(T(n, k), k=0..n), n=0..9);

# uses[bell_matrix from A264428]
bell_matrix(lambda n: 2 if n<2 else 0, 12) # Peter Luschny, Jan 19 2016

Defining relation: 2*x*(2*x-1)*...*(2*x-n+1) = sum {k=0..n} T(n, k)*x*(x-1)*...*(x-k+1)
Explicit formula: T(n,k) = (n!/k!)*binomial(k,n-k)*2^(2*k-n). [As defined by Comtet (see reference).]
Recurrence relation: T(n,k) = (2*k-n+1)*T(n-1,k)+2*T(n-1,k-1).
E.g.f.: exp(x*(t^2+2*t)) = 1 + (2*x)*t + (2*x+4*x^2)*t^2/2! + (12*x^2+8*x^3)*t^3/3! + ...
O.g.f. for m-th diagonal (starting at main diagonal m = 0): (2*m)!/m!*x^m/(1-2*x)^(2*m+1).
The triangle is the matrix product [2^k*s(n,k)]n,k>=0 * ([s(n,k)]n,k>=0)^(-1),
where s(n,k) denotes the signed Stirling number of the first kind.
Row sums are [1,2,6,20,76,...] = A000898.
Column sums are [1,4,28,296,...] = [2^n*A001515(n)] n>=0.

A305536 Expansion of 1/(1 - x/(1 - x - 1x/(1 - x - 2x/(1 - x - 3x/(1 - x - 4x/(1 - ...)))))), a continued fraction.

Original entry on oeis.org

1, 1, 3, 12, 62, 410, 3426, 35360, 438390, 6358306, 105544388, 1970997142, 40860191470, 930482058472, 23079257369054, 619157277351618, 17860295754328884, 551188620179519302, 18119420989759583998, 632069815329176122584, 23318435171385786420958, 907077442499274638005314

Offset: 0

Ilya Gutkovskiy, Jun 04 2018

nonn

Invert transform of A001515, shifted right one place.

Alois P. Heinz, Table of n, a(n) for n = 0..405
N. J. A. Sloane, Transforms
Index entries for sequences related to Bessel functions or polynomials

Cf. A001003, A001515, A144301, A258173.

b:= proc(n) option remember;
     `if`(n<2, n+1, (2*n-1)*b(n-1)+b(n-2))
    end:
a:= proc(n) option remember;
     `if`(n=0, 1, add(b(j-1)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 11 2023

nmax = 21; CoefficientList[Series[1/(1 - x/(1 - x + ContinuedFractionK[-k x, 1 - x, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 21; CoefficientList[Series[1/(1 - Sum[HypergeometricPFQ[{k, 1 - k}, {}, -1/2] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[HypergeometricPFQ[{k, 1 - k}, {}, -1/2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

a(n) ~ 2^(n - 1/2) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Sep 18 2021

A305537 G.f. A(x) satisfies: A(x) = 1/(1 - xA(x) - xA(x)/(1 - xA(x) - 2xA(x)/(1 - xA(x) - 3xA(x)/(1 - xA(x) - 4x*A(x)/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 2, 11, 87, 844, 9438, 118217, 1636078, 24869591, 414422424, 7568815758, 151468591827, 3317061005044, 79265498450882, 2058189152006115, 57777549430984983, 1744191365957251044, 56332730020388347302, 1937412176139535240463, 70659708678402399722656

Offset: 0

Ilya Gutkovskiy, Jun 04 2018

nonn

			G.f. A(x) = 1 + 2*x + 11*x^2 + 87*x^3 + 844*x^4 + 9438*x^5 + 118217*x^6 + 1636078*x^7 + 24869591*x^8 + ...

Index entries for sequences related to Bessel functions or polynomials

Cf. A001515, A027307, A301363.

a(n) = [x^n] (Sum_{k>=0} A001515(k)*x^k)^(n+1)/(n + 1).

cp's OEIS Frontend

A368726 Number of non-isomorphic connected multiset partitions of weight n into singletons or pairs.

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A368727 Number of non-isomorphic connected multiset partitions of weight n into singletons or strict pairs.

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Author

Keywords

Examples

Links

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Programs

Mathematica

Formula

A144575 E,g.f.: exp(1-sqrt(1-2*x-3*x^2)).

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Mathematica

Formula

A144905 a(0) = 1; thereafter a(n) = A105749(n)/n.

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Maple

A144906 a(0) = 1; thereafter a(n) = A144422(n)/n.

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A185296 Triangle of connection constants between the falling factorials (x)(n) and (2*x)(n).

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Maple

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A305536 Expansion of 1/(1 - x/(1 - x - 1*x/(1 - x - 2*x/(1 - x - 3*x/(1 - x - 4*x/(1 - ...)))))), a continued fraction.

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A305537 G.f. A(x) satisfies: A(x) = 1/(1 - x*A(x) - x*A(x)/(1 - x*A(x) - 2*x*A(x)/(1 - x*A(x) - 3*x*A(x)/(1 - x*A(x) - 4*x*A(x)/(1 - ...))))), a continued fraction.

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A144575 E,g.f.: exp(1-sqrt(1-2x-3x^2)).

A305536 Expansion of 1/(1 - x/(1 - x - 1x/(1 - x - 2x/(1 - x - 3x/(1 - x - 4x/(1 - ...)))))), a continued fraction.

A305537 G.f. A(x) satisfies: A(x) = 1/(1 - xA(x) - xA(x)/(1 - xA(x) - 2xA(x)/(1 - xA(x) - 3xA(x)/(1 - xA(x) - 4x*A(x)/(1 - ...))))), a continued fraction.