A129306 -id:A129306 - cp's OEIS frontend

A138533 Resort the multinomial sequence A036038 by source partition as described in A126442, A129306 and A136101.

1, 2, 1, 6, 3, 1, 24, 12, 4, 1, 6, 120, 60, 20, 5, 1, 30, 10, 720, 360, 120, 30, 6, 1, 180, 60, 15, 20, 90

Offset: 1

Alford Arnold, Mar 27 2008

Multinomials count permutations of multisets and also paths in lattices; for example, there are six paths (from null to full) through the lattice of divisors for signature 36: 2233 2323 2332 3223 3232 and 3322.

			a(11) is six because the eleventh least prime signature in source format is 36 the signature for partition 2+2 the ninth partition and A036038(9) = 6.
The tables begin:
1.......2.......6.......24......120.....720....5040.....40320......362880
........1.......3.......12.......60.....360....2520.....20160......181440
................1.......4........20.....120.....840......6720.......60480
........................1........5.......30.....210......1680.......15120
.. ..............................1........6......42......336........3024
..........................................1.......7.......56.........504
..................................................1........8..........72
...........................................................1...........9
.......................................................................1
........................6........30.....180....1260....10080........90720
.................................10......60.....420.....3360........30240
...

Cf. A000142 A001710 A005651 (column sums) A025487 A053445.

Cf. A173333. [From Reinhard Zumkeller, Feb 19 2010]

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

A346426 Number A(n,k) of partitions of the (n+k)-multiset {0,...,0,1,2,...,k} with n 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 5, 4, 3, 15, 15, 11, 7, 5, 52, 52, 36, 21, 12, 7, 203, 203, 135, 74, 38, 19, 11, 877, 877, 566, 296, 141, 64, 30, 15, 4140, 4140, 2610, 1315, 592, 250, 105, 45, 22, 21147, 21147, 13082, 6393, 2752, 1098, 426, 165, 67, 30, 115975, 115975, 70631, 33645, 13960, 5317, 1940, 696, 254, 97, 42

Offset: 0

Alois P. Heinz, Jul 16 2021

Also number A(n,k) of factorizations of 2^n * Product_{i=1..k} prime(i+1); A(3,1) = 7: 2*2*2*3, 2*3*4, 4*6, 2*2*6, 3*8, 2*12, 24; A(1,2) = 5: 2*3*5, 5*6, 3*10, 2*15, 30.

			A(2,2) = 11: 00|1|2, 001|2, 1|002, 0|0|1|2, 0|01|2, 0|1|02, 01|02, 00|12, 0|0|12, 0|012, 0012.
Square array A(n,k) begins:
   1,  1,   2,    5,   15,    52,    203,     877,    4140, ...
   1,  2,   5,   15,   52,   203,    877,    4140,   21147, ...
   2,  4,  11,   36,  135,   566,   2610,   13082,   70631, ...
   3,  7,  21,   74,  296,  1315,   6393,   33645,  190085, ...
   5, 12,  38,  141,  592,  2752,  13960,   76464,  448603, ...
   7, 19,  64,  250, 1098,  5317,  28009,  158926,  963913, ...
  11, 30, 105,  426, 1940,  9722,  52902,  309546, 1933171, ...
  15, 45, 165,  696, 3281, 16972,  95129,  572402, 3670878, ...
  22, 67, 254, 1106, 5372, 28582, 164528, 1015356, 6670707, ...
  ...

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

Columns k=0-10 give: A000041, A000070, A082775, A093802, A346857, A346858, A346859, A346860, A346861, A346862, A346863.
Rows n=0+1, 2-10 give: A000110, A035098, A169587, A169588, A346851, A346852, A346853, A346854, A346855, A346856.
Main diagonal gives A346424.
Antidiagonal sums give A346428.
Cf. A000079, A001055, A008277, A048993, A070826, A126442, A129306, A219727, A255903, A292508, A346520.

s:= proc(n) option remember; expand(`if`(n=0, 1,
      x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
S:= proc(n, k) option remember; coeff(s(n), x, k) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=0,
      combinat[numbpart](n), add(b(n-j, i-1), j=0..n)))
    end:
A:= (n, k)-> add(S(k, j)*b(n, j), j=0..k):
seq(seq(A(n, d-n), n=0..d), d=0..10);

s[n_] := s[n] = Expand[If[n == 0, 1, x Sum[s[n - j] Binomial[n - 1, j - 1], {j, 1, n}]]];
S[n_, k_] := S[n, k] = Coefficient[s[n], x, k];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, PartitionsP[n], Sum[b[n - j, i - 1], {j, 0, n}]]];
A[n_, k_] := Sum[S[k, j] b[n, j], {j, 0, k}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Aug 18 2021, after Alois P. Heinz *)

A(n,k) = A001055(A000079(n)*A070826(k+1)).
A(n,k) = Sum_{j=0..k} A048993(k,j)*A292508(n,j+1).
A(n,k) = Sum_{j=0..k} Stirling2(k,j)*Sum_{i=0..n} binomial(j+i-1,i)*A000041(n-i).

A126442 Triangular array t read by rows: t(0,k) is p(k), the number of partitions of the k-multiset {0,0,...,0} with k zeros. For 0 <= n < k, t(n, k) is the number of partitions of the k-multiset {0, 0, ..., 0, 1, 2, 3, ..., k-n} with n zeros.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 5, 7, 11, 15, 7, 12, 21, 36, 52, 11, 19, 38, 74, 135, 203, 15, 30, 64, 141, 296, 566, 877, 22, 45, 105, 250, 592, 1315, 2610, 4140, 30, 67, 165, 426, 1098, 2752, 6393, 13082, 21147, 42, 97, 254, 696, 1940, 5317, 13960, 33645, 70631, 115975

Offset: 1

Alford Arnold, Jan 28 2007

First in a series of triangular arrays which comprise subsequences of A096443(n).
The second array begins 9 16 26 29 52 92 47 98 198 371 and when the arrays are aligned as illustrated in triangle A126441 with p(n) values they sum to A035310 which counts unordered multisets.
Let t(n, k) be the number of ways to partition the k-multiset {0,0,...,0,1,2,3,4,...,k-n} with n zeros, 0 <= n < k. Then t(n, k) = sum_i = 0..k j = 0..n S(n, j) C(i,  j) p(k - n - i), where S(n, j) are Stirling numbers of the second kind, C(i, j) are the number of compositions of i distinct objects into j parts, and p is the integer partition function.
To see this, partition [n] into j blocks; there are S(n, j) partitions. For such a partition x and for each i, there are C(i, j) ways to distribute i zeros into x, because the blocks of x are all distinct. There are p(k-n-i) ways to partition the remaining k-n-i zeros. Multiplying and summing gives the result. - George Beck, Jan 10 2011
Values are also part of A096443, A129306 and A249620. Columns are also columns of the last one of these irregular triangles. See "Partitions_of_multisets" link. - Tilman Piesk, Nov 09 2014

			This first array includes only the hook cases. A096443(9,14,16) correspond to partitions [2,2], [3,2] and [2,2,1] so these values do not appear in A126442.
The array begins:
1
2 2
3 4 5
5 7 11 15
7 12 21 36 52

Tilman Piesk, Partitions of multisets (Wikiversity)

Cf. A000041, A000070, A082775, A093802, A000291, A002763, A000412, A054225, A035310, A000110, A035098.

(* The triangle is flattened to a sequence. *)
t[n_, k_] := Sum[StirlingS2[n, j] * Binomial[-1 + i + j, i] * PartitionsP[k - n - i], {j, 0, n}, {i, 0, k - n}]; Table[ t[n, k], {k, 10}, {n, 0, k - 1}] // Flatten (* George Beck, Jan 10 2011 *)

Definition clarified by George Beck, Jan 11 2011

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).

Original entry on oeis.org

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

A002763 Number of bipartite partitions.

Original entry on oeis.org

4, 11, 26, 52, 98, 171, 289, 467, 737, 1131, 1704, 2515, 3661, 5246, 7430, 10396, 14405, 19760, 26884, 36269, 48583, 64614, 85399, 112170, 146526, 190362, 246099, 316621, 405556, 517224, 657012, 831320, 1048055, 1316611, 1648486, 2057324, 2559719, 3175309

Offset: 0

N. J. A. Sloane

nonn

M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 11.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..300
M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956 (Annotated scanned pages from, plus a review)

Cf. A082775, A129306.

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((45*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, DeleteCases[Divisors[n], 1|n]}]]; a[n_] := b[45*2^n, 45*2^n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 20 2014, after Alois P. Heinz *)
nmax = 100; CoefficientList[Series[(4 - x - 3*x^2 + x^3) / ((1 - x)^3 * (1 + x)) / Product[1 - x^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 07 2017 *)

a(n) = a(n-1) + A000041(n) + A000070(n) + A000291(n), for n>0 - Alford Arnold, Dec 10 2007
From Vaclav Kotesovec, Jan 07 2017: (Start)
G.f.: (4 - x - 3*x^2 + x^3) / ((1-x)^3 * (1+x)) * Product_{k>=1} 1/(1-x^k).
a(n) ~ exp(Pi*sqrt(2*n/3)) * 3*sqrt(n)/(2*sqrt(2)*Pi^3).
(End)

Extended beyond a(25) by Alois P. Heinz, May 26 2013

A136101 Resort sequence A025487 by source partition as described by A053445 and A126442.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 30, 16, 24, 60, 210, 36, 32, 48, 120, 420, 2310, 72, 180, 64, 96, 240, 840, 4620, 30030, 144, 360, 1260, 216, 900, 128, 192, 480, 1680, 9240, 60060, 510510, 288, 720, 2520, 13860, 432, 1080, 1800, 6300, 256, 384, 960, 3360, 18480, 120120

Offset: 0

Alford Arnold, Jan 01 2008

			Let a(0) = 1 then let the irregular table begin:
2....4....8....16....32....64....
....6...12....24....48....96....
........30....60...120...240....
.............210...420...840....
..................2310..4620....
.......................30030....
..............36....72...144....
...................180...360....
........................1260....
.........................216....
.........................900....

Cf. A025487 A053445 A126442 A129306.

A131419 Central diagonal and left half of table A054225 which counts the partitions of n objects of 2 colors.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 7, 9, 7, 12, 16, 11, 19, 29, 31, 15, 30, 47, 57, 22, 45, 77, 97, 109, 30, 67, 118, 162, 189, 42, 97, 181, 257, 323, 339, 56, 139, 267, 401, 522, 589, 77, 195, 392, 608, 831, 975, 1043, 101, 272, 560, 907, 1279, 1576, 1752, 135, 373

Offset: 0

Alford Arnold, Jul 11 2007

Every diagonal appears in sequence A129306.

			Table A054225 begins
1
1...1
2...2...2
3...4...4...3
5...7...9...7...5
7..12..16..16..12..7
The diagonals in the new table begin
1..1..2..3..5...7...11...15...22...30..42 A000041
.....2..4..7..12...19...30...45...67..97 A000070
........9.16..29...47...77..118..181
..........31..57...97..162..257
.............109..189..323
..................339

Cf. A108464, A129306.

More terms from R. J. Mathar, Sep 02 2007

A131888 Follow each least prime signature by the corresponding value in the array A096443.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 2, 8, 3, 12, 4, 30, 5, 16, 5, 24, 7, 36, 9, 60, 11, 210, 15, 32, 7, 48, 12, 72, 16, 120, 21, 180, 26, 420, 36, 2310, 52, 64, 11, 96, 19, 144, 29, 240, 38, 216, 31, 360, 52, 840, 74, 900, 66, 1260, 92, 4620, 135, 30030, 203

Offset: 0

Alford Arnold, Jul 23 2007

			Array A096443 begins
1..1..2..3...5...7..11..15...22...30
.....2..4...7..12..19..30...45...67
............9..16..29..47...77..118
........5..11..21..38..64..105..165
etc
so the sequence begins 1 1 2 1 4 2 6 2 8 3 12 4 30 5 ...

Cf. A001055 A025487 A080577 A096443 A129306 A131822.

cp's OEIS Frontend

A138533 Resort the multinomial sequence A036038 by source partition as described in A126442, A129306 and A136101.

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

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A346426 Number A(n,k) of partitions of the (n+k)-multiset {0,...,0,1,2,...,k} with n 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A126442 Triangular array t read by rows: t(0,k) is p(k), the number of partitions of the k-multiset {0,0,...,0} with k zeros. For 0 <= n < k, t(n, k) is the number of partitions of the k-multiset {0, 0, ..., 0, 1, 2, 3, ..., k-n} with n zeros.

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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

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A002763 Number of bipartite partitions.

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A136101 Resort sequence A025487 by source partition as described by A053445 and A126442.

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A131419 Central diagonal and left half of table A054225 which counts the partitions of n objects of 2 colors.

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