A035098 -id:A035098 - cp's OEIS frontend

A249620 Triangle read by rows: T(m,n) = number of partitions of the multiset with m elements and signature corresponding to n-th integer partition (A194602).

1, 1, 2, 2, 5, 4, 3, 15, 11, 7, 9, 5, 52, 36, 21, 26, 12, 16, 7, 203, 135, 74, 92, 38, 52, 19, 66, 29, 31, 11, 877, 566, 296, 371, 141, 198, 64, 249, 98, 109, 30, 137, 47, 57, 15, 4140, 2610, 1315, 1663, 592, 850, 250, 1075, 392, 444, 105, 560

Offset: 0

Tilman Piesk, Nov 04 2014

This triangle shows the same numbers in each row as A129306 and A096443, but in this arrangement the multisets in column n correspond to the n-th integer partition in the infinite order defined by A194602.
Row lengths: A000041 (partition numbers), Row sums: A035310
Columns: 0: A000110 (Bell), 1: A035098 (near-Bell), 2: A169587, 4: A169588
Last in row: end-1: A091437, end: A000041 (partition numbers)
The rightmost columns form a reflected version of the triangle A126442:
  n      0     1     2     4    6   10  14  21      (A000041(1,2,3...)-1)
m
1        1
2        2     2
3        5     4     3
4       15    11     7     5
5       52    36    21    12    7
6      203   135    74    38   19   11
7      877   566   296   141   64   30  15
8     4140  2610  1315   592  250  105  45  22
A249619 shows the number of permutations of the same multisets.

			See "The T(5,2)=21 partitions of {1,1,1,2,3}" link. Similar links for m=1..8 are in "Partitions of multisets" (Wikiversity).
Triangle begins:
  n     0    1   2   3   4   5   6   7   8   9  10
m
0       1
1       1
2       2    2
3       5    4   3
4      15   11   7   9   5
5      52   36  21  26  12  16   7
6     203  135  74  92  38  52  19  66  29  31  11

Tilman Piesk, Triangle rows m=0..8, flattened.
Tilman Piesk, Partitions of multisets (Wikiversity)
Tilman Piesk, The T(5,2)=21 partitions of {1,1,1,2,3}
Tilman Piesk, PHP code used to calculate the examples

Cf. A129306, A096443, A035310, A194602, A249619, A000041, A126442

A093802 Number of distinct factorizations of 105*2^n.

Original entry on oeis.org

5, 15, 36, 74, 141, 250, 426, 696, 1106, 1711, 2593, 3852, 5635, 8118, 11548, 16231, 22577, 31092, 42447, 57464, 77213, 103009, 136529, 179830, 235514, 306751, 397506, 512607, 658030, 841020, 1070490, 1357195, 1714274, 2157539, 2706174, 3383187, 4216358

Offset: 0

Alford Arnold, May 19 2004

			105*A000079 is 105, 210, 420, 840, 1680, 3360, ... and there are 15 distinct factorizations of 210 so a(1) = 15.
a(0) = 5: 105*2^0 = 105 = 3*5*7 = 3*35 = 5*21 = 7*15. - _Alois P. Heinz_, May 26 2013

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Similar sequences: 45*A000079 => A002763, [1, 3, 9, 27, 81, 243...]*A000079 => A054225, 1*A002110 => A000110, 2*A002110 => A035098, A000142 => A076716.
Cf. A001055, A126442, A129306, A346823.
Column k=3 of A346426.

with(numtheory):
b:= proc(n, k) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
    end:
a:= n-> b((105*2^n)$2):
seq(a(n), n=0..50);  # Alois P. Heinz, May 26 2013

b[n_, k_] := b[n, k] = If[n > k, 0, 1] + If[PrimeQ[n], 0,
     Sum[If[d > k, 0, b[n/d, d]], {d, Divisors[n][[2;;-2]]}]];
a[n_] := b[105*2^n, 105*2^n];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 15 2021, after Alois P. Heinz *)

2 more terms from Alford Arnold, Aug 29 2007

Corrected offset and extended beyond a(7) by Alois P. Heinz, May 26 2013

A169587 The total number of ways of partitioning the multiset {1,1,1,2,3,...,n-2}.

Original entry on oeis.org

3, 7, 21, 74, 296, 1315, 6393, 33645, 190085, 1145246, 7318338, 49376293, 350384315, 2606467211, 20266981269, 164306340566, 1385709542808, 12133083103491, 110095025916745, 1033601910417425, 10024991744613469, 100316367530768074, 1034373400144455266

Offset: 3

Martin Griffiths, Dec 02 2009

nonn

			The partitions of {1,1,1,2} are {{1},{1},{1},{2}}, {{1,1},{1},{2}}, {{1,2},{1},{1}}, {{1,1},{1,2}}, {{1,1,1},{2}}, {{1,1,2},{1}} and {{1,1,1,2}}, so a(4)=7.

Alois P. Heinz, Table of n, a(n) for n = 3..576
M. Griffiths, I. Mezo, A generalization of Stirling Numbers of the Second Kind via a special multiset, JIS 13 (2010) #10.2.5.
M. Griffiths, Generalized Near-Bell Numbers, JIS 12 (2009) 09.5.7

This is related to A000110, A035098 and A169588.
Row n=3 of A346426.
Cf. A346813.

Table[(BellB[n] + 3 BellB[n - 1] + 5 BellB[n - 2] + 2 BellB[n - 3])/ 6, {n, 3, 23}]

For n>=3, a(n)=(Bell(n)+3Bell(n-1)+5Bell(n-2)+2Bell(n-3))/6, where Bell(n) is the n-th Bell number (the Bell numbers are given in A000110).

E.g.f.: (e^(3x)+6e^(2x)+9e^x+2)(e^(e^x-1))/6.

A169588 The total number of ways of partitioning the multiset {1,1,1,1,2,3,...,n-3}.

Original entry on oeis.org

5, 12, 38, 141, 592, 2752, 13960, 76464, 448603, 2801054, 18516832, 129034659, 944356507, 7235605732, 57879020756, 482189616711, 4174720731316, 37489711726834, 348592657600818, 3350919079643612, 33252861484374737, 340209759518479300, 3584240435109146792

Offset: 4

Martin Griffiths, Dec 02 2009

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..576
M. Griffiths, I. Mezo, A generalization of Stirling Numbers of the Second Kind via a special multiset, JIS 13 (2010) #10.2.5.
M. Griffiths, Generalized Near-Bell Numbers, JIS 12 (2009) 09.5.7

This is related to A000110, A035098 and A169587.
Row n=4 of A346426.
Cf. A346814.

Table[(BellB[n] + 6 BellB[n - 1] + 17 BellB[n - 2] + 20 BellB[n - 3] + 21 BellB[n - 4])/24, {n, 4, 23}]

For n>=4, a(n)=(Bell(n)+6Bell(n-1)+17Bell(n-2)+20Bell(n-3)+21Bell(n-4))/24, where Bell(n) is the n-th Bell number (the Bell numbers are given in A000110). e.g.f. (e^(4x)+12e^(3x)+42e^(2x)+44e^x+21)(e^(e^x-1))/24.

A241500 Triangle T(n,k): number of ways of partitioning the n-element multiset {1,1,2,3,...,n-1} into exactly k nonempty parts, n>=1 and 1<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 4, 1, 1, 11, 16, 7, 1, 1, 23, 58, 41, 11, 1, 1, 47, 196, 215, 90, 16, 1, 1, 95, 634, 1041, 640, 176, 22, 1, 1, 191, 1996, 4767, 4151, 1631, 315, 29, 1, 1, 383, 6178, 21001, 25221, 13587, 3696, 526, 37, 1, 1, 767, 18916, 90055, 146140, 105042, 38409, 7638, 831, 46, 1

Offset: 1

Andrew Woods, Apr 24 2014

			There are 58 ways to partition {1,1,2,3,4,5} into three nonempty parts.
The first few rows are:
  1;
  1,   1;
  1,   2,    1;
  1,   5,    4,     1;
  1,  11,   16,     7,     1;
  1,  23,   58,    41,    11,     1;
  1,  47,  196,   215,    90,    16,    1;
  1,  95,  634,  1041,   640,   176,   22,   1;
  1, 191, 1996,  4767,  4151,  1631,  315,  29,  1;
  1, 383, 6178, 21001, 25221, 13587, 3696, 526, 37, 1;
  ...

The first five columns appear as A000012, A083329, A168583, A168584, A168585.

Row sums give A035098.

T(n,k) = stirling(n-1,k,2) + stirling(n-1,k-1,2) + binomial(k,2)*stirling(n-2,k,2); \\ Michel Marcus, Apr 24 2014

T(n,k) = S(n-1,k) + S(n-1,k-1) + C(k,2)*S(n-2,k), where S refers to Stirling numbers of the second kind (A008277), and C to binomial coefficients (A007318).

A093936 Table T(n,k) read by rows which contains in row n and column k the sum of A001055(A036035(n,j)) over all column indices j where A036035(n,j) has k distinct prime factors.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 5, 16, 11, 15, 7, 28, 47, 36, 52, 11, 79, 156, 166, 135, 203, 15, 134, 408, 588, 667, 566, 877, 22, 328, 1057, 2358, 2517, 2978, 2610, 4140, 30, 536, 3036, 6181, 10726, 11913, 14548, 13082, 21147, 42, 1197, 6826, 21336, 40130, 53690, 61421

Offset: 1

Alford Arnold, May 23 2004

Sequence A050322 calculates factorizations indexed by prime signatures: A001055(A025487) For example, A050322(36) = A001055(A025487(36)) = 74 and A050322(43) = A001055(A024487(43)) = 92.

Note that A093936 can be readily extended by combining appropriate values from A096443. Row sums of A093936 yield A035310 and embedded sequences include A000041, A035098 and A000110. - Alford Arnold, Nov 19 2005

			a(19) = 166 because A001055(840) + A001055(1260) = 74 + 92.
Row n=4 of A036035 contains 16=2^4, 24=2^3*3, 36=2^2*3^2, 60=2^2*3*5 and 210=2*3*5*7. The 16 has k=1 distinct prime factor; 24 and 36 have k=2 distinct prime factors; 60 has k=3 distinct prime factors; 210 has k=4 distinct prime factors (see A001221).
T(4,1)=A001055(16)=5.
T(4,2)=A001055(24)+A001055(36)=7+9=16.
T(4,3)=A001055(60)=11.
T(4,4)=A001055(210)=15.
Table starts
1;
2, 2;
3, 4, 5;
5, 16, 11, 15;
7, 28, 47, 36, 52;
11, 79, 156, 166, 135, 203;
15, 134, 408, 588, 667, 566, 877;
22, 328, 1057, 2358, 2517, 2978, 2610, 4140;
30, 536, 3036, 6181, 10726, 11913, 14548, 13082, 21147;
42, 1197, 6826, 21336, 40130, 53690, 61421, 76908, 70631, 115975;
...

Cf. A001055, A050322.

Cf. A000041, A000110, A035098, A035310, A096443.

A036035 := proc(n) local pr,L,a ; a := [] ; pr := combinat[partition](n) ; for L in pr do mul(ithprime(i)^op(-i,L),i=1..nops(L)) ; a := [op(a),%] ; od ; RETURN(a) ; end: A001221 := proc(n) local ifacts ; ifacts := ifactors(n)[2] ; nops(ifacts) ; end: listProdRep := proc(n,mincomp) local dvs,resul,f,i,rli ; resul := 0 ; if n = 1 then RETURN(1) elif n >= mincomp then dvs := numtheory[divisors](n) ; for i from 1 to nops(dvs) do f := op(i,dvs) ; if f =n and f >= mincomp then resul := resul+1 ; elif f >= mincomp then rli := listProdRep(n/f,f) ; resul := resul+rli ; fi ; od ; fi ; RETURN(resul) ; end: A001055 := proc(n) listProdRep(n,2) ; end: A093936 := proc(n,k) local a, a036035,j ; a := 0 ; a036035 := A036035(n) ; for j in a036035 do if A001221(j) = k then a := a+A001055(j) ; fi ; od ; RETURN(a) ; end: for n from 1 to 10 do for k from 1 to n do printf("%d,",A093936(n,k)) ; od : od : # R. J. Mathar, Jul 27 2007

More terms from Alford Arnold, Nov 19 2005

More terms from R. J. Mathar, Jul 27 2007

A087649 a(n) = (1/2)*(Bell(n+2)-Bell(n+1)+Bell(n)).

Original entry on oeis.org

1, 2, 6, 21, 83, 363, 1733, 8942, 49484, 291871, 1825501, 12054705, 83734241, 609851830, 4644041462, 36883843101, 304846039251, 2616765134351, 23286746418237, 214489200063218, 2041785040262972, 20060079966396887, 203156789589084133, 2118391734395139205

Offset: 0

Vladeta Jovovic, Sep 23 2003

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Reinis Cirpons, James East, and James D. Mitchell, Transformation representations of diagram monoids, arXiv:2411.14693 [math.RA], 2024. See pp. 3, 33.

Cf. A000110, A035098, A059606.

[(1/2)*(Bell(n+2)-Bell(n+1)+Bell(n)): n in [0..30]]; // Vincenzo Librandi, Nov 13 2011

G.f.: (1-x+x^2)/(2*x*Q(0)) - 1/(2*x) + 1/2, where Q(k)= 1 - x - x/(1 - x*(2*k+1)/(1 - x - x/(1 - x*(2*k+2)/Q(k+1)))); (continued fraction). - Sergei N. Gladkovskii, May 13 2013

E.g.f.: exp(exp(x) - 1) * (exp(2*x) + 1) / 2. - Ilya Gutkovskiy, Aug 09 2021

A131420 A tabular sequence of arrays counting ordered factorizations over least prime signatures. The unordered version is described by sequence A129306.

Original entry on oeis.org

1, 2, 3, 4, 8, 13, 8, 20, 44, 75, 26, 16, 48, 132, 308, 541, 76, 176, 32, 112, 368, 1076, 2612, 4683, 208, 604, 1460, 252, 818, 64, 256, 876, 3408, 10404, 25988, 47293, 544, 1888, 5740, 14300, 768, 2316, 3172, 7880, 128, 576, 2496, 10096, 36848, 116180

Offset: 1

Alford Arnold, Jul 10 2007

nonn

The display has 1 2 3 5 7 11 15 ... terms per column. (cf. A000041)
The arrays begin
1.....2.....4......8......16.....32.....64......128
......3.....8.....20......48....112....256......576
...........13.....44.....132....368....976.....2496
..................75.....308...1076...3408....10096
.........................541...2612..10404....36848
...............................4683..25988...116180
.....................................47293...296564
.............................................545835
..................26......76....208....544
.........................176....604...1888
...............................1460...5740
.....................................14300
................................252....768
......................................2316
................................818...3172
......................................7880
with column sums
1....5....25....173....1297....12225....124997 => A035341
Column i corresponds to partitions of i. The rows correspond successively to the partitions {i}, {i-1,1},{i-2,1,1},{i-3,1,1,1}, ..., {i-7,1,1,1,1,1,1,1}, {i-2,2}, {i-3,2,1}, {i-4,2,1,1}, {i-5,2,1,1,1}, {i-3,3}, {i-3,3,1}, {i-4,2,2}, {i-5,2,2,1}. - Roger Lipsett, Feb 26 2016

			36 = 2*2*3*3 and is in A025487. There are 26 ways to factor 36 so a(11) = 26.

Cf. A000041, A000670, A002033, A025487, A035098, A035341, A050324, A074206, A095705, A098348, A104725, A108464, A129306.

gozinta counts ordered factorizations of an integer, and if lst is a partition we have
gozinta[1] = 1;
gozinta[n_] := gozinta[n] = 1 + Sum[gozinta[n/i], {i, Rest@Most@Divisors@n}]
a[lst_] := gozinta[Times @@ (Array[Prime, Length@lst]^lst)] (* Roger Lipsett, Feb 26 2016 *)

Corrected entries in table in comments section - Roger Lipsett, Feb 26 2016

cp's OEIS Frontend

A249620 Triangle read by rows: T(m,n) = number of partitions of the multiset with m elements and signature corresponding to n-th integer partition (A194602).

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A093802 Number of distinct factorizations of 105*2^n.

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Maple

Mathematica

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A169587 The total number of ways of partitioning the multiset {1,1,1,2,3,...,n-2}.

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Mathematica

Formula

A169588 The total number of ways of partitioning the multiset {1,1,1,1,2,3,...,n-3}.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A241500 Triangle T(n,k): number of ways of partitioning the n-element multiset {1,1,2,3,...,n-1} into exactly k nonempty parts, n>=1 and 1<=k<=n.

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PARI

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A093936 Table T(n,k) read by rows which contains in row n and column k the sum of A001055(A036035(n,j)) over all column indices j where A036035(n,j) has k distinct prime factors.

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Maple

Extensions

A087649 a(n) = (1/2)*(Bell(n+2)-Bell(n+1)+Bell(n)).

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Magma

Formula

A131420 A tabular sequence of arrays counting ordered factorizations over least prime signatures. The unordered version is described by sequence A129306.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions