A079025 -id:A079025 - cp's OEIS frontend

A122172 Triangle read by rows relating A074139, A074141, A078436 and A079025.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 4, 3, 1, 1, 4, 6, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 3, 3, 2, 1, 1, 3, 4, 4, 3, 1, 1, 3, 5, 5, 3, 1, 1, 4, 7, 7, 4, 1, 1, 5, 10, 10, 5, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Alford Arnold, Aug 23 2006

A proper definition is needed for this sequence.
Are the row sums A074139(n) and the row lengths A000041(n)? - R. J. Mathar, May 08 2019 [Not exactly: see below. - M. F. Hasler, Jan 07 2024]
From M. F. Hasler, Jan 06 2024: (Start)
I get this triangle as T(n,k) = # { v in S(p_n), |v| = k }, where p_n is the n-th partition as listed in A036036 or A036037 (which has a nice table of the p's), and S(p) = {0, ..., p[1]} x ... x {0, ..., p[#p]}, the set of vectors v with 0 <= v[i] <= p[i] for all indices i from 1 to #p = number of parts in p.
Then the row sums are indeed the total number of elements in S(p_n) which is equal to the product (p[1]+1)*...*(p[#p]+1) which is also the number of divisors of the Heinz number of p (cf. A185974).
The row lengths are 1 + |p| = 1 + sum of all parts of p (corresponding to the possible values of |v| ranging from 0 to |p|), repeated A000041(|p|) times: A000041(0) = 1 row of length 0+1 for the partition () of 0, A000041(1) = 1 row of length 1+1 for partition (1) of 1; A000041(2) = 2 rows of length 2+1 for the two partitions (2) and (1,1) of 2;  A000041(3) = 3 rows of length 3+1 for the 3 partitions {(3), (2,1), (1,1,1)} of 3; etc. (End)

			The triangle begins:
  1
  1 1
  1 1 1
  1 2 1
  1 1 1 1
  1 2 2 1
  1 3 3 1
  1 1 1 1 1
  1 2 2 2 1
  1 2 3 2 1
  1 3 4 3 1
  1 4 6 4 1
  1 1 1 1 1 1
  1 2 2 2 2 1
  1 2 3 3 2 1
  1 3 4 4 3 1
  1 3 5 5 3 1
  1 4 7 7 4 1
  1 5 10 10 5 1

M. F. Hasler, Table of n, a(n) for n = 0..9172 (up to partitions of 15), Jan 07 2024

Cf. A036036 (partitions in A-S order), A036037 (the same, parts reversed), A185974 (corresponding Heinz numbers).

A122172_row(n, p=part(n))={my(c=Vec(0, vecsum(p)+1)); forvec(v=[[0, k]| k<-p], c[vecsum(v)+1]++); c} \\ instead of n one can directly give p as 2nd arg
/* helper function: n-th partition as listed in A036036, A036037 or A185974 */
part(n)={my(c, r=0); while(n >= c = numbpart(r), n -= c; r++); partitions(r)[n+1]}
for(n=0,5, forpart(p=n, print(A122172_row(, Vec(p))) )) \\ Illustration. \\ M. F. Hasler, Jan 06 2024

More terms from M. F. Hasler, Jan 07 2024

A122453 Triangular array related to A122402 as A079025 relates to A122172; sums A079139 values associated with cyclic partitions.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 3, 2, 2, 3, 4, 4, 3, 2, 4, 8, 13, 15, 13, 8, 4, 4, 8, 13, 16, 16, 13, 8, 4, 7, 17, 32, 45, 51, 45, 32, 17, 7, 8, 20, 38, 56, 67, 67, 56, 38, 20, 8, 12, 34, 72, 117, 156, 171, 156, 117, 72, 34, 12, 14, 41, 88, 147, 203, 237, 237, 203, 147, 88, 41

Offset: 0

Alford Arnold, Sep 07 2006

			The table begins:
1
0 0
1 1 1
1 1 1 1
2 3 4 3 2
2 3 4 4 3 2
4 8 13 15 13 8 4
4 8 13 16 16 13 8 4
7 17 32 45 51 45 32 17 7
8 20 38 56 67 67 56 38 20 8
12 34 72 117 156 171 156 117 72 34 12
14 41 88 147 203 237 237 203 147 88 41 14

Cf. A002865 A074139 A079025 A122172 A122402.

A074141 Sum of products of parts increased by 1 in all partitions of n.

Original entry on oeis.org

1, 2, 7, 18, 50, 118, 301, 684, 1621, 3620, 8193, 17846, 39359, 84198, 181313, 383208, 811546, 1695062, 3546634, 7341288, 15207022, 31261006, 64255264, 131317012, 268336125, 545858260, 1110092387, 2250057282, 4558875555, 9213251118, 18613373708, 37529713890

Offset: 0

Amarnath Murthy, Aug 28 2002

nonn

Replace each term in A036035 by the number of its divisors as in A074139; sequence gives sum of terms in the n-th row.

This is the sum of the number of submultisets of the multisets with n elements; a part of a partition is a frequency of such an element. - George Beck, Nov 01 2011

			The partitions of 4 are 4, 3+1, 2+2, 2+1+1, 1+1+1+1, the corresponding products when parts are increased by 1 are 5,8,9,12,16 and their sum is a(4) = 50.

Alois P. Heinz, Table of n, a(n) for n = 0..3317

Cf. A006906, A036035, A074140, A256155.
Row sums of A074139 and of A079025 and of A079308 and of A238963.
Column k=2 of A261718.
Cf. A267008.

b:= proc(n, i) option remember; `if`(n=0 or i=1,
      2^n, b(n, i-1) +(1+i)*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50); # Alois P. Heinz, Sep 07 2014

Table[Plus @@ Times @@@ (IntegerPartitions[n] + 1), {n, 0, 28}] (* T. D. Noe, Nov 01 2011 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, (1+i) * b[n-i, i]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 08 2015, after Alois P. Heinz *)

S(n,m):=if n=0 then 1 else if nVladimir Kruchinin, Sep 07 2014 */

G.f.: 1/Product_{m>0} (1-(m+1)*x^m).
a(n) = 1/n*Sum_{k=1..n} b(k)*a(n-k), where b(k) = Sum_{d divides k} d*(d+1)^(k/d).
a(n) = S(n,1), where S(n,m) = sum(k=m..n/2, (k+1)*S(n-k,k))+(n+1), S(n,n)=n+1, S(0,m)=1, S(n,m)=0 for nVladimir Kruchinin, Sep 07 2014
a(n) ~ c * 2^n, where c = Product_{k>=2} 1/(1-(k+1)/2^k) = 18.56314656361011472747535423226928404842588594722907068201... = A256155. - Vaclav Kotesovec, Sep 11 2014, updated May 10 2021

More terms from Alford Arnold, Sep 17 2002

More terms, better description and formulas from Vladeta Jovovic, Vladimir Baltic, Nov 28 2002

A309691 Total number of divisors d of m (counted with multiplicity), such that the prime signature of d is a partition of two and m runs through the set of least numbers whose prime signature is a partition of n.

Original entry on oeis.org

2, 6, 16, 32, 65, 113, 199, 323, 523, 803, 1237, 1826, 2696, 3873, 5544, 7767, 10859, 14912, 20425, 27598, 37156, 49473, 65679, 86350, 113179, 147191, 190806, 245676, 315415, 402522, 512250, 648551, 818831, 1029139, 1290151, 1610701, 2005969, 2489167, 3081434

Offset: 2

Alois P. Heinz, Aug 23 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..7500

Column k=2 of A079025.

A309693 Total number of divisors d of m (counted with multiplicity), such that the prime signature of d is a partition of three and m runs through the set of least numbers whose prime signature is a partition of n.

Original entry on oeis.org

3, 12, 32, 79, 160, 318, 573, 1013, 1683, 2776, 4366, 6820, 10325, 15503, 22721, 33105, 47289, 67177, 93990, 130747, 179636, 245613, 332243, 447368, 597142, 793508, 1046512, 1374713, 1793842, 2332053, 3014392, 3882511, 4975306, 6354950, 8079980, 10241877

Offset: 3

Alois P. Heinz, Aug 23 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..4000

Column k=3 of A079025.

A309915 Total number of divisors d of m (counted with multiplicity), such that the prime signature of d is a partition of n and m runs through the set of least numbers whose prime signature is a partition of 2n.

Original entry on oeis.org

1, 3, 16, 79, 371, 1683, 7413, 31769, 133692, 553848, 2265776, 9181670, 36928673, 147650125, 587734595, 2331625130, 9226486717, 36443758767, 143763811785, 566624864014, 2232055573265, 8789903797692, 34610963678036, 136287108614677, 536724439657635

Offset: 0

Alois P. Heinz, Aug 22 2019

nonn

			a(2) = 16: The partitions of 2*2 are (4), (31), (22), (211), (1111). Least numbers with these prime signatures are 16, 24, 36, 60, 210. Their divisors with prime signatures (2) or (11) are {4}, {4,6}, {4,6,9}, {4,6,10,15}, {6,10,14,15,21,35}. The total number is 1 + 2 + 3 + 4 + 6.

Alois P. Heinz, Table of n, a(n) for n = 0..380

Cf. A000041, A079025, A087443.

b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, (x+1)^n,
      b(n, i-1) +factor((x^(i+1)-1)/(x-1))*b(n-i, min(n-i, i))))
    end:
a:= n-> coeff(b(2*n$2), x, n):
seq(a(n), n=0..25);

b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, (x + 1)^n, b[n, i - 1] + Factor[(x^(i + 1) - 1)/(x - 1)] b[n - i, Min[n - i, i]]]];
a[n_] := Coefficient[b[2n, 2n], x, n];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 12 2020, after Alois P. Heinz *)

a(n) = A079025(2n,n).

A309919 Total number of divisors d of m (counted with multiplicity), such that the prime signature of d is a partition of four and m runs through the set of least numbers whose prime signature is a partition of n.

Original entry on oeis.org

5, 20, 65, 160, 371, 756, 1485, 2701, 4822, 8144, 13558, 21756, 34449, 53044, 80874, 120573, 178215, 258897, 373069, 530068, 748026, 1043001, 1445321, 1983124, 2705411, 3659491, 4924840, 6578409, 8746089, 11553224, 15194682, 19870115, 25881012, 33537548

Offset: 4

Alois P. Heinz, Aug 23 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..4000

Column k=4 of A079025.

A309920 Total number of divisors d of m (counted with multiplicity), such that the prime signature of d is a partition of five and m runs through the set of least numbers whose prime signature is a partition of n.

Original entry on oeis.org

7, 35, 113, 318, 756, 1683, 3405, 6662, 12205, 21840, 37415, 62866, 102248, 163920, 256132, 395419, 598461, 896112, 1320378, 1928165, 2777751, 3970068, 5610839, 7872981, 10941361, 15109772, 20691740, 28174131, 38084160, 51210142, 68419073, 90977599, 120279526

Offset: 5

Alois P. Heinz, Aug 23 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..4000

Column k=5 of A079025.

A309921 Total number of divisors d of m (counted with multiplicity), such that the prime signature of d is a partition of six and m runs through the set of least numbers whose prime signature is a partition of n.

Original entry on oeis.org

11, 54, 199, 573, 1485, 3405, 7413, 14901, 28943, 53365, 95822, 165612, 280689, 461838, 747905, 1183801, 1848372, 2833288, 4294109, 6408566, 9470481, 13817889, 19985430, 28592582, 40594736, 57092122, 79745667, 110478215, 152106462, 207912108, 282607366

Offset: 6

Alois P. Heinz, Aug 23 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..4000

Column k=6 of A079025.

A309922 Total number of divisors d of m (counted with multiplicity), such that the prime signature of d is a partition of seven and m runs through the set of least numbers whose prime signature is a partition of n.

Original entry on oeis.org

15, 86, 323, 1013, 2701, 6662, 14901, 31769, 63626, 123049, 227506, 410058, 714139, 1218779, 2025892, 3310989, 5298045, 8360139, 12962789, 19860393, 29990209, 44815019, 66147220, 96748294, 140005126, 200971214, 285833838, 403589570, 565249022, 786529323

Offset: 7

Alois P. Heinz, Aug 23 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..4000

Column k=7 of A079025.

cp's OEIS Frontend

A122172 Triangle read by rows relating A074139, A074141, A078436 and A079025.

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A122453 Triangular array related to A122402 as A079025 relates to A122172; sums A079139 values associated with cyclic partitions.

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A074141 Sum of products of parts increased by 1 in all partitions of n.

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A309691 Total number of divisors d of m (counted with multiplicity), such that the prime signature of d is a partition of two and m runs through the set of least numbers whose prime signature is a partition of n.

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A309693 Total number of divisors d of m (counted with multiplicity), such that the prime signature of d is a partition of three and m runs through the set of least numbers whose prime signature is a partition of n.

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A309915 Total number of divisors d of m (counted with multiplicity), such that the prime signature of d is a partition of n and m runs through the set of least numbers whose prime signature is a partition of 2n.

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A309919 Total number of divisors d of m (counted with multiplicity), such that the prime signature of d is a partition of four and m runs through the set of least numbers whose prime signature is a partition of n.

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A309920 Total number of divisors d of m (counted with multiplicity), such that the prime signature of d is a partition of five and m runs through the set of least numbers whose prime signature is a partition of n.

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A309921 Total number of divisors d of m (counted with multiplicity), such that the prime signature of d is a partition of six and m runs through the set of least numbers whose prime signature is a partition of n.

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A309922 Total number of divisors d of m (counted with multiplicity), such that the prime signature of d is a partition of seven and m runs through the set of least numbers whose prime signature is a partition of n.

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