A346517 -id:A346517 - cp's OEIS frontend

A094574 Number of (<=2)-covers of an n-set.

1, 1, 5, 40, 457, 6995, 136771, 3299218, 95668354, 3268445951, 129468914524, 5868774803537, 301122189141524, 17327463910351045, 1109375488487304027, 78484513540137938209, 6098627708074641312182, 517736625823888411991202, 47791900951140948275632148

Offset: 0

Goran Kilibarda, Vladeta Jovovic, May 12 2004

nonn

Also the number of strict multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}. For example, the a(2) = 5 strict multiset partitions of {1, 1, 2, 2} are (1122), (1)(122), (2)(112), (11)(22), (1)(2)(12). - Gus Wiseman, Jul 18 2018

			From _Gus Wiseman_, Sep 02 2019: (Start)
These are set-systems covering {1..n} with vertex-degrees <= 2. For example, the a(3) = 40 covers are:
  {123}  {1}{23}    {1}{2}{3}     {1}{2}{3}{12}
         {2}{13}    {1}{2}{13}    {1}{2}{3}{13}
         {3}{12}    {1}{2}{23}    {1}{2}{3}{23}
         {1}{123}   {1}{3}{12}    {1}{2}{13}{23}
         {12}{13}   {1}{3}{23}    {1}{2}{3}{123}
         {12}{23}   {2}{3}{12}    {1}{3}{12}{23}
         {13}{23}   {2}{3}{13}    {2}{3}{12}{13}
         {2}{123}   {1}{12}{23}
         {3}{123}   {1}{13}{23}
         {12}{123}  {1}{2}{123}
         {13}{123}  {1}{3}{123}
         {23}{123}  {2}{12}{13}
                    {2}{13}{23}
                    {2}{3}{123}
                    {3}{12}{13}
                    {3}{12}{23}
                    {12}{13}{23}
                    {1}{23}{123}
                    {2}{13}{123}
                    {3}{12}{123}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..300

Row n=2 of A219585. - Alois P. Heinz, Nov 23 2012
Dominated by A003465.
Graphs with vertex-degrees <= 2 are A136281.
Cf. A002718, A007716, A020554, A020555, A050535, A094574, A136284, A316974, A327104, A327106, A327229.
Main diagonal of A346517.

facs[n_]:=facs[n]=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[Array[Prime,n,1,Times]^2],UnsameQ@@#&]],{n,0,6}] (* Gus Wiseman, Jul 18 2018 *)
m = 20;
a094577[n_] := Sum[Binomial[n, k]*BellB[2 n - k], {k, 0, n}];
egf = Exp[(1 - Exp[x])/2]*Sum[a094577[n]*(x/2)^n/n!, {n, 0, m}] + O[x]^m;
CoefficientList[egf + O[x]^m, x]*Range[0, m-1]! (* Jean-François Alcover, May 13 2019 *)

Row sums of A094573.

E.g.f: exp(-1-1/2*(exp(x)-1))*Sum(exp(x*binomial(n+1, 2))/n!, n=0..infinity) or exp((1-exp(x))/2)*Sum(A094577 (n)*(x/2)^n/n!, n=0..infinity).

A346500 Number A(n,k) of partitions of the (n+k)-multiset {1,2,...,n,1,2,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 4, 4, 5, 15, 11, 9, 11, 15, 52, 36, 26, 26, 36, 52, 203, 135, 92, 66, 92, 135, 203, 877, 566, 371, 249, 249, 371, 566, 877, 4140, 2610, 1663, 1075, 712, 1075, 1663, 2610, 4140, 21147, 13082, 8155, 5133, 3274, 3274, 5133, 8155, 13082, 21147

Offset: 0

Alois P. Heinz, Jul 20 2021

Also number A(n,k) of factorizations of Product_{i=1..n} prime(i) * Product_{i=1..k} prime(i); A(2,2) = 9: 2*2*3*3, 3*3*4, 6*6, 2*3*6, 4*9, 2*2*9, 3*12, 2*18, 36.

			A(2,2) = 9: 1122, 11|22, 12|12, 1|122, 112|2, 11|2|2, 1|1|22, 1|12|2, 1|1|2|2.
Square array A(n,k) begins:
    1,    1,    2,     5,    15,     52,     203,     877, ...
    1,    2,    4,    11,    36,    135,     566,    2610, ...
    2,    4,    9,    26,    92,    371,    1663,    8155, ...
    5,   11,   26,    66,   249,   1075,    5133,   26683, ...
   15,   36,   92,   249,   712,   3274,   16601,   91226, ...
   52,  135,  371,  1075,  3274,  10457,   56135,  325269, ...
  203,  566, 1663,  5133, 16601,  56135,  198091, 1207433, ...
  877, 2610, 8155, 26683, 91226, 325269, 1207433, 4659138, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns (or rows) k=0-10 give: A000110, A035098, A322764, A322768, A346881, A346882, A346883, A346884, A346885, A346886, A346887.
Main diagonal gives A020555.
First upper (or lower) diagonal gives A322766.
Second upper (or lower) diagonal gives A322767.
Antidiagonal sums give A346490.
A(2n,n) gives A322769.
Cf. A001055, A002110, A322765, A346426, A346517.

g:= proc(n, k) option remember; uses numtheory; `if`(n>k, 0, 1)+
     `if`(isprime(n), 0, add(`if`(d>k or max(factorset(n/d))>d, 0,
        g(n/d, d)), d=divisors(n) minus {1, n}))
    end:
p:= proc(n) option remember; `if`(n=0, 1, p(n-1)*ithprime(n)) end:
A:= (n, k)-> g(p(n)*p(k)$2):
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
A:= proc(n, k) option remember; `if`(n

b[n_] := b[n] = If[n == 0, 1, Sum[b[n-j] Binomial[n-1, j-1], {j, 1, n}]];
A[n_, k_] := A[n, k] = If[n < k, A[k, n],
     If[k == 0, b[n], (A[n + 1, k - 1] + Sum[A[n - k + j, j]*
     Binomial[k - 1, j], {j, 0, k - 1}] + A[n, k - 1])/2]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Aug 18 2021, after Alois P. Heinz's second program *)

A(n,k) = A001055(A002110(n)*A002110(k)).
A(n,k) = A(k,n).
A(n,k) = A322765(abs(n-k),min(n,k)).

A322773 Column 2 of array in A322770.

Original entry on oeis.org

5, 18, 70, 299, 1393, 7023, 38043, 220054, 1352082, 8784991, 60125371, 432001747, 3248914361, 25508188118, 208592396802, 1772921926183, 15632838989393, 142759592985079, 1348095912827295, 13145321614286610, 132188675368994446, 1369216940917868547

Offset: 0

N. J. A. Sloane, Dec 30 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..572

Cf. A322764, A322770.

Column k=2 of A346517.

A322774 Column 3 of array in A322770.

Original entry on oeis.org

40, 172, 801, 4025, 21709, 124997, 764538, 4945866, 33710579, 241273791, 1807949285, 14146621349, 115316563400, 977216138500, 8592652709041, 78263082518169, 737228862573509, 7172071557558805, 71964172085006666, 743866850349092130, 7912230538914051723

Offset: 0

N. J. A. Sloane, Dec 30 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..570

Cf. A322768, A322770.

Column k=3 of A346517.

A322771 Row 1 of array in A322770.

Original entry on oeis.org

1, 3, 18, 172, 2295, 40043, 875936, 23308546, 737478487, 27252363585, 1159335917625, 56103737161197, 3057787510932485, 186102920689311261, 12555513437042340449, 932964243520888524391, 75926403820972271325522, 6733532223196893844825456, 647846856775383975668238328, 67349524752043243630964385000

Offset: 0

N. J. A. Sloane, Dec 30 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A322770, A346517.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
A:= proc(n, k) option remember; `if`(n A(n, n+1):
seq(a(n), n=0..19);  # Alois P. Heinz, Jul 21 2021

Q[m_, n_] := Q[m, n] = If[n == 0, BellB[m], (1/2)(Q[m+2, n-1] + Q[m+1, n-1] - Sum[Binomial[n-1, k] Q[m, k], {k, 0, n-1}])];
a[n_] := Q[1, n];
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Apr 29 2022 *)

a(n) = A346517(n,n+1) = A346517(n+1,n). - Alois P. Heinz, Jul 21 2021

A322772 Row 2 of array in A322770.

Original entry on oeis.org

2, 9, 70, 801, 12347, 243235, 5908978, 172449180, 5925731200, 235946129714, 10745098631229, 553630279110396, 31978001903989065, 2054387367168242251, 145795148420558536232, 11361381129471379493270, 967044630942570464100761, 89483154423059719127570924, 8963545185499520505954151682

Offset: 0

N. J. A. Sloane, Dec 30 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A322770, A346517.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
A:= proc(n, k) option remember; `if`(n A(n, n+2):
seq(a(n), n=0..18);  # Alois P. Heinz, Jul 21 2021

Q[m_, n_] := Q[m, n] = If[n == 0, BellB[m], (1/2)(Q[m+2, n-1] + Q[m+1, n-1] - Sum[Binomial[n-1, k] Q[m, k], {k, 0, n-1}])];
a[n_] := Q[2, n];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Apr 29 2022 *)

a(n) = A346517(n,n+2) = A346517(n+2,n). - Alois P. Heinz, Jul 21 2021

A346518 Total number of partitions of all n-multisets {1,2,...,n-j,1,2,...,j} into distinct multisets for 0 <= j <= n.

Original entry on oeis.org

1, 2, 5, 16, 53, 202, 826, 3724, 17939, 93390, 516125, 3042412, 18923139, 124368810, 857827458, 6208594458, 46937360868, 370335617694, 3039823038753, 25928519847988, 229285625745624, 2099543718917418, 19872430464012935, 194203934113959970, 1956736801957704866

Offset: 0

Alois P. Heinz, Jul 21 2021

nonn

Also total number of factorizations of Product_{i=1..n-j} prime(i) * Product_{i=1..j} prime(i) into distinct factors for 0 <= j <= n.

Alois P. Heinz, Table of n, a(n) for n = 0..576

Antidiagonal sums of A346517.

Cf. A002110, A045778, A346490.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
A:= proc(n, k) option remember; `if`(n add(A(n-j, j), j=0..n):
seq(a(n), n=0..24);

(* Q is A322770 *)
Q[m_, n_] := Q[m, n] = If[n == 0, BellB[m], (1/2) (Q[m + 2, n - 1] +
     Q[m + 1, n - 1] - Sum[Binomial[n - 1, k] Q[m, k], {k, 0, n - 1}])];
A[n_, k_] := Q[Abs[n - k], Min[n, k]];
a[n_] := Sum[A[n - j, j], {j, 0, n}];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Apr 06 2022 *)

a(n) = Sum_{j=0..n} A045778(A002110(n-j)*A002110(j)).

a(n) = Sum_{j=0..n} A346517(n-j,j).

A346897 Number of partitions of the (n+4)-multiset {1,2,...,n,1,2,3,4} into distinct multisets.

Original entry on oeis.org

15, 31, 70, 172, 457, 2295, 12347, 70843, 431636, 2781372, 18885177, 134670425, 1005626319, 7842880313, 63733789544, 538521313772, 4722239916109, 42899823779887, 403127907995427, 3912789622898471, 39175487045290456, 404110470650751416, 4290001467563698869

Offset: 0

Alois P. Heinz, Aug 06 2021

nonn

Also number of factorizations of Product_{i=1..n} prime(i) * Product_{i=1..4} prime(i) into distinct factors.

Alois P. Heinz, Table of n, a(n) for n = 0..572

Column (or row) k=4 of A346517.

Cf. A346881.

A346898 Number of partitions of the (n+5)-multiset {1,2,...,n,1,2,...,5} into distinct multisets.

Original entry on oeis.org

52, 120, 299, 801, 2295, 6995, 40043, 243235, 1562071, 10569612, 75114998, 559057663, 4346341361, 35213909313, 296696424975, 2594712611306, 23512150424424, 220412149622759, 2134476336680003, 21324751529837231, 219524646452224135, 2325946109640859828

Offset: 0

Alois P. Heinz, Aug 06 2021

nonn

Also number of factorizations of Product_{i=1..n} prime(i) * Product_{i=1..5} prime(i) into distinct factors.

Alois P. Heinz, Table of n, a(n) for n = 0..572

Column (or row) k=5 of A346517.

Cf. A346882.

A346899 Number of partitions of the (n+6)-multiset {1,2,...,n,1,2,...,6} into distinct multisets.

Original entry on oeis.org

203, 514, 1393, 4025, 12347, 40043, 136771, 875936, 5908978, 41862462, 310606617, 2407527172, 19449364539, 163419459850, 1425408016084, 12884224833364, 120496112842571, 1164268932225644, 11607009161248659, 119245195695018868, 1261021161600404210

Offset: 0

Alois P. Heinz, Aug 06 2021

nonn

Also number of factorizations of Product_{i=1..n} prime(i) * Product_{i=1..6} prime(i) into distinct factors.

Alois P. Heinz, Table of n, a(n) for n = 0..571

Column (or row) k=6 of A346517.

Cf. A346883.

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A094574 Number of (<=2)-covers of an n-set.

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A346500 Number A(n,k) of partitions of the (n+k)-multiset {1,2,...,n,1,2,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A322773 Column 2 of array in A322770.

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A322774 Column 3 of array in A322770.

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A322771 Row 1 of array in A322770.

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A322772 Row 2 of array in A322770.

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A346518 Total number of partitions of all n-multisets {1,2,...,n-j,1,2,...,j} into distinct multisets for 0 <= j <= n.

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A346897 Number of partitions of the (n+4)-multiset {1,2,...,n,1,2,3,4} into distinct multisets.

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A346898 Number of partitions of the (n+5)-multiset {1,2,...,n,1,2,...,5} into distinct multisets.

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