A066186 -id:A066186 - cp's OEIS frontend

A340061 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(n-m) copies of m, with n >= 1 and m >= 1.

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

Offset: 1

Omar E. Pol, Dec 28 2020

Conjecture: all divisors of all terms of row n are also all parts of all partitions of n.
The conjecture gives a correspondence between divisors and partitions (see example).
It is conjectured that every section of the set of partitions of n has essentially the same correspondence. For more information see A336811.

			Triangle begins:
  1;
  1, 2;
  1, 1, 2, 3;
  1, 1, 1, 2, 2, 3, 4;
  1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5;
  1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6;
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, ...
  ...
For n = 6 the 6th row of the triangle consists of:
  p(5) = 7 copies of 1, that is, [1, 1, 1, 1, 1, 1, 1],
  p(4) = 5 copies of 2, that is, [2, 2, 2, 2, 2],
  p(3) = 3 copies of 3, that is, [3, 3, 3],
  p(2) = 2 copies of 4, that is, [4, 4],
  p(1) = 1 copy   of 5, that is, [5],
  p(0) = 1 copy   of 6, that is, [6],
where p(j) is the j-th partition number A000041(j).
About the conjecture we have that the divisors of the terms of the 6th row are:
                                                                     1
                                                            1, 1,    2
                                    1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 3
  6th row -->  1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6
There are nineteen 1's, eight 2's, four 3's, two 4's, one 5 and one 6.
In total there are 19 + 8 + 4 + 2 + 1 + 1 = 35 divisors.
On the other hand the partitions of 6 are:
      Diagram          Parts
    _ _ _ _ _ _
   |_ _ _      |       6
   |_ _ _|_    |       3 3
   |_ _    |   |       4 2
   |_ _|_ _|_  |       2 2 2
   |_ _ _    | |       5 1
   |_ _ _|_  | |       3 2 1
   |_ _    | | |       4 1 1
   |_ _|_  | | |       2 2 1 1
   |_ _  | | | |       3 1 1 1
   |_  | | | | |       2 1 1 1 1
   |_|_|_|_|_|_|       1 1 1 1 1 1
There are nineteen 1's, eight 2's, four 3's, two 4's, one 5 and one 6, as shown also the 6th row of A066633.
In total there are 19 + 8 + 4 + 2 + 1 + 1 = A006128(6) = 35 parts.
In accordance with the conjecture we can see that all divisors of all terms of the 6th row of triangle are the same positive integers as all parts of all partitions of 6.

Paolo Xausa, Table of n, a(n) for n = 1..10980 (rows 1..21 of the triangle, flattened)

Mirror of A176206.
Row sums give A014153.
Row n has length A000070(n-1).
Right border gives A000027.
Cf. A000041, A006128, A066633, A027750, A066186, A176206, A336811 (section).

A340061row[n_]:=Flatten[Table[ConstantArray[m,PartitionsP[n-m]],{m,n}]];Array[A340061row,10] (* Paolo Xausa, Sep 01 2023 *)

A340793 Sequence whose partial sums give A000203.

Original entry on oeis.org

1, 2, 1, 3, -1, 6, -4, 7, -2, 5, -6, 16, -14, 10, 0, 7, -13, 21, -19, 22, -10, 4, -12, 36, -29, 11, -2, 16, -26, 42, -40, 31, -15, 6, -6, 43, -53, 22, -4, 34, -48, 54, -52, 40, -6, -6, -24, 76, -67, 36, -21, 26, -44, 66, -48, 48, -40, 10, -30, 108, -106, 34, 8

Offset: 1

Omar E. Pol, Jan 21 2021

Essentially a duplicate of A053222.
Convolved with the nonzero terms of A000217 gives A175254, the volume of the stepped pyramid described in A245092.
Convolved with the nonzero terms of A046092 gives A244050, the volume of the stepped pyramid described in A244050.
Convolved with A000027 gives A024916.
Convolved with A000041 gives A138879.
Convolved with A000070 gives the nonzero terms of A066186.
Convolved with the nonzero terms of A002088 gives A086733.
Convolved with A014153 gives A182738.
Convolved with A024916 gives A000385.
Convolved with A036469 gives the nonzero terms of A277029.
Convolved with A091360 gives A276432.
Convolved with A143128 gives the nonzero terms of A000441.
For the correspondence between divisors and partitions see A336811.

1 together with A053222.
Cf. A000203 (partial sums).
Cf. A000027, A000041, A000070, A000217, A000385, A000441, A002088, A002961, A014153, A024916, A036469, A046092, A053223, A053224, A053225, A053226, A053227, A066186, A086733, A091360, A138879, A143128, A175254, A182738, A237593, A244050, A245092, A276432, A277029, A336811.

a:= n-> (s-> s(n)-s(n-1))(numtheory[sigma]):
seq(a(n), n=1..77);  # Alois P. Heinz, Jan 21 2021

Join[{1}, Differences @ Table[DivisorSigma[1, n], {n, 1, 100}]] (* Amiram Eldar, Jan 21 2021 *)

a(n) = if (n==1, 1, sigma(n)-sigma(n-1)); \\ Michel Marcus, Jan 22 2021

a(n) = A053222(n-1) for n>1. - Michel Marcus, Jan 22 2021

A343342 Number of integer partitions of n with no part dividing or divisible by all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 3, 2, 5, 5, 12, 7, 22, 20, 32, 34, 60, 54, 98, 93, 145, 159, 237, 229, 361, 384, 529, 574, 810, 840, 1194, 1275, 1703, 1886, 2484, 2660, 3566, 3909, 4987, 5520, 7092, 7737, 9907, 10917, 13603, 15226, 18910, 20801, 25912, 28797

Offset: 0

Gus Wiseman, Apr 15 2021

nonn

Alternative name: Number of integer partitions of n that are either empty or have smallest part not dividing all the others and greatest part not divisible by all the others.

			The a(0) = 1 through a(12) = 7 partitions (empty columns indicated by dots):
  ()  .  .  .  .  (32)  .  (43)   (53)   (54)    (64)    (65)     (75)
                           (52)   (332)  (72)    (73)    (74)     (543)
                           (322)         (432)   (433)   (83)     (552)
                                         (522)   (532)   (92)     (732)
                                         (3222)  (3322)  (443)    (4332)
                                                         (533)    (5322)
                                                         (542)    (33222)
                                                         (722)
                                                         (3332)
                                                         (4322)
                                                         (5222)
                                                         (32222)

The opposite version is A130714.
The first condition alone gives A338470.
The Heinz numbers of these partitions are A343338 = A342193 /\ A343337.
The second condition alone gives A343341.
The half-opposite versions are A343344 and A343345.
The "or" instead of "and" version is A343346 (strict: A343382).
The strict case is A343379.
A000009 counts strict partitions.
A000041 counts partitions.
A000070 counts partitions with a selected part (strict: A015723).
Cf. A006128, A066186, A083710, A130689, A341450, A343343, A343377.

Table[Length[Select[IntegerPartitions[n],#=={}||!And@@IntegerQ/@(#/Min@@#)&&!And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]

A066967 Total sum of odd parts in all partitions of n.

Original entry on oeis.org

1, 2, 7, 10, 23, 36, 65, 94, 160, 230, 356, 502, 743, 1030, 1480, 2006, 2797, 3760, 5120, 6780, 9092, 11902, 15701, 20350, 26508, 34036, 43860, 55822, 71215, 89988, 113792, 142724, 179137, 223230, 278183, 344602, 426687, 525616, 647085, 792950

Offset: 1

Vladeta Jovovic, Jan 26 2002

nonn

Partial sums of A206435. - Omar E. Pol, Mar 17 2012
From Omar E. Pol, Apr 01 2023: (Start)
Convolution of A000041 and A000593.
Convolution of A002865 and A078471.
a(n) is also the sum of all odd divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned odd divisors are also all odd parts of all partitions of n. (End)

			a(4) = 10 because in the partitions of 4, namely [4],[3,1],[2,2],[2,1,1],[1,1,1,1], the total sum of the odd parts is (3+1)+(1+1)+(1+1+1+1) = 10.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
George E. Andrews and Mircea Merca, A further look at the sum of the parts with the same parity in the partitions of n, Journal of Combinatorial Theory, Series A, Volume 203, 105849 (2024).

Cf. A000041, A000593, A066897, A066898, A113685, A206435.

Cf. A002865, A078471.

g:=sum((2*i-1)*x^(2*i-1)/(1-x^(2*i-1)),i=1..50)/product(1-x^j,j=1..50): gser:=series(g,x=0,50): seq(coeff(gser,x^n),n=1..47);
# Emeric Deutsch, Feb 19 2006
b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, n]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+ (i mod 2)*g[1]*i]
      fi
    end:
a:= n-> b(n, n)[2]:
seq (a(n), n=1..50);
# Alois P. Heinz, Mar 22 2012

max = 50; g = Sum[(2*i-1)*x^(2*i-1)/(1-x^(2*i-1)), {i, 1, max}]/Product[1-x^j, {j, 1, max}]; gser = Series[g, {x, 0, max}]; a[n_] := SeriesCoefficient[gser, {x, 0, n}]; Table[a[n], {n, 1, max-1}] (* Jean-François Alcover, Jan 24 2014, after Emeric Deutsch *)
Map[Total[Select[Flatten[IntegerPartitions[#]], OddQ]] &, Range[30]] (* Peter J. C. Moses, Mar 14 2014 *)

a(n) = Sum_{k=1..n} b(k)*numbpart(n-k), where b(k)=A000593(k)=sum of odd divisors of k.
a(n) = sum(k*A113685(n,k), k=0..n). - Emeric Deutsch, Feb 19 2006
G.f.: sum((2i-1)x^(2i-1)/(1-x^(2i-1)), i=1..infinity)/product(1-x^j, j=1..infinity). - Emeric Deutsch, Feb 19 2006
a(n) = A066186(n) - A066966(n). - Omar E. Pol, Mar 10 2012
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)). - Vaclav Kotesovec, May 29 2018

More terms from Naohiro Nomoto and Sascha Kurz, Feb 07 2002

A220909 The second crank moment function M_2(n).

Original entry on oeis.org

0, 2, 8, 18, 40, 70, 132, 210, 352, 540, 840, 1232, 1848, 2626, 3780, 5280, 7392, 10098, 13860, 18620, 25080, 33264, 44088, 57730, 75600, 97900, 126672, 162540, 208208, 264770, 336240, 424204, 534336, 669438, 837080, 1041810, 1294344, 1601138, 1977140, 2432430, 2987040, 3655806

Offset: 0

N. J. A. Sloane, Jan 02 2013

nonn

M_2(n) is defined to be Sum_{m=-n..n} m^2 M(m,n) where M(m,n) is the number of partitions of n with crank m except for n=1 where M(-1,1) = M(1,1) = -M(0,1) = 1. - Michael Somos, Nov 10 2013
From Omar E. Pol, Jul 25 2022: (Start)
Apart from the initial zero this is also:
Convolution of A074400 and A000041.
Convolution of A000203 and A139582. (End)

			G.f. = 2*x + 8*x^2 + 18*x^3 + 40*x^4 + 70*x^5 + 132*x^6 + 210*x^7 + ...
For n=1, M_2(1) = Sum_{m=-1..1} m^2 * M(m,2) = (-1)^2*1 + 0^2*(-1) + 1^2*1 = 2. For n=2, the partition [2] has crank 2 and partition [1,1] has crank -2, hence M_2(2) = 2^2 + (-2)^2 = 8. - _Michael Somos_, Nov 10 2013

G. C. Greubel, Table of n, a(n) for n = 0..5000
F. G. Garvan, Higher-order spt functions, arXiv:1008.1207 [math.NT], 2010.
F. G. Garvan, Higher-order spt functions, Adv. Math. 228 (2011), no. 1, 241-265.
Wikipedia, Crank of a partition

Cf. A000041, A066186, A092269, A220908.

Cf. A000203, A074400, A139582.

a[ n_] := 2 n PartitionsP @ n (* Michael Somos, Nov 10 2013 *)

{a(n) = if( n<0, 0, 2 * n * polcoeff( 1 / eta(x + x * O(x^n)), n))} /* Michael Somos, Nov 10 2013 */

a(n) = 2*n*A000041(n) = 2*A066186(n).
a(n) = n*A139582(n). - Omar E. Pol, Jan 03 2013
a(n) = A220908(n) + A211982(n), n >= 1. - Omar E. Pol, Jan 17 2013
a(n) = 2*(A092269(n) + A220907(n)), n >= 1. _Omar E. Pol, Feb 18 2013
a(n) ~ exp(Pi*sqrt(2*n/3))/(2*sqrt(3)) * (1 - (sqrt(3/2)/Pi + Pi/(24*sqrt(6))) / sqrt(n)). - Vaclav Kotesovec, Oct 24 2016

A221649 Tetrahedron E(n,j,k) = kT(j,k)p(n-j), where T(j,k) = 1 if k divides j otherwise 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 1, 0, 3, 3, 2, 4, 1, 0, 3, 1, 2, 0, 4, 5, 3, 6, 2, 0, 6, 1, 2, 0, 4, 1, 0, 0, 0, 5, 7, 5, 10, 3, 0, 9, 2, 4, 0, 8, 1, 0, 0, 0, 5, 1, 2, 3, 0, 0, 6, 11, 7, 14, 5, 0, 15, 3, 6, 0, 12, 2, 0, 0, 0, 10, 1, 2, 3, 0, 0, 6, 1, 0, 0, 0, 0, 0, 7

Offset: 1

Omar E. Pol, Jan 21 2013

The tetrahedron shows a connection between divisors and partitions.
The sum of all elements of slice n is A066186(n).
The sum of row j of slice n is A221529(n,j).
The sum of column k of slice n is A138785(n,k), the sum of all parts of size k in all partitions of n.
See also the tetrahedron of A221650.

			First five slices of tetrahedron are
---------------------------------------------------
n  j / k   1  2  3  4  5  6      A221529   A066186
---------------------------------------------------
1  1       1,                       1         1
...................................................
2  1       1,                       1
2  2       1, 2,                    3         4
...................................................
3  1       2,                       2
3  2       1, 2,                    3
3  3       1, 0, 3,                 4         9
...................................................
4  1       3,                       3
4  2       2, 4,                    6
4  3       1, 0, 3,                 4
4  4       1, 2, 0, 4,              7        20
...................................................
5  1       5,                       5
5  2       3, 6,                    9
5, 3,      2, 0, 6,                 8
5, 4,      1, 2, 0, 4,              7
5, 5,      1, 0, 0, 0, 5,           6        35
...................................................
.
From _Omar E. Pol_, Jul 26 2021: (Start)
The slices of the tetrahedron appear in the upper zone of the following table (formed by four zones) which shows the correspondence between divisors and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n |         |  1  |   2   |    3    |     4     |      5      |
|---|---------|-----|-------|---------|-----------|-------------|
|   |    -    |     |       |         |           |  5          |
| C |    -    |     |       |         |  3        |  3 6        |
| O |    -    |     |       |  2      |  2 4      |  2 0 6      |
| N | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
| D | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
| D | A127093 |     |       |         |           |  1          |
| I |---------|-----|-------|---------|-----------|-------------|
| V | A127093 |     |       |         |  1        |  1 2        |
| I | A127093 |     |       |         |  1        |  1 2        |
| S | A127093 |     |       |         |  1        |  1 2        |
| O |---------|-----|-------|---------|-----------|-------------|
| R | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| S | A127093 |     |       |  1      |  1 2      |  1 0 3      |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
| L | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
| I |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| N | A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| K |         |  |  |  |\|  |  |\|\|  |  |\|\|\|  |  |\|\|\|\|  |
|   | A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
| P |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
| A |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| R |         |     |       |  3      |  3 1      |  3 1 1      |
| T |         |     |       |         |  2 2      |  2 2 1      |
| I |         |     |       |         |  4        |  4 1        |
| T |         |     |       |         |           |  3 2        |
| I |         |     |       |         |           |  5          |
| O |         |     |       |         |           |             |
| N |         |     |       |         |           |             |
| S |         |     |       |         |           |             |
|---|---------|-----|-------|---------|-----------|-------------|
.
The upper zone is a condensed version of the "divisors" zone.
The above table is the table of A340011 upside down.
For more information about the correspondence divisor/part see A338156. (End)

Paolo Xausa, Table of n, a(n) for n = 1..11480 (rows n = 1..40 of the tetrahedron, flattened)

Nonzero terms give A340057.

Cf. A000005, A000041, A000203, A027750, A051731, A066186, A127093, A138785, A221529, A221650, A237593, A336811, A336812, A338156, A340011, A340031, A340032, A340035, A340056.

A221649row[n_]:=Flatten[Table[If[Divisible[j,k],PartitionsP[n-j]k,0],{j,n},{k,j}]];Array[A221649row,10] (* Paolo Xausa, Sep 26 2023 *)

E(n,j,k) = k*A051731(j,k)*A000041(n-j) = A127093(j,k)*A000041(n-j) = k*A221650(n,j,k).

a(18)-a(19) and a(28)-a(29) corrected by Paolo Xausa, Sep 26 2023

A222730 Total sum T(n,k) of parts <= n of multiplicity k in all partitions of n; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

0, 0, 1, 3, 2, 1, 11, 6, 0, 1, 36, 10, 3, 0, 1, 79, 21, 3, 1, 0, 1, 186, 33, 7, 3, 1, 0, 1, 345, 59, 9, 4, 1, 1, 0, 1, 672, 89, 20, 4, 4, 1, 1, 0, 1, 1163, 145, 22, 11, 4, 2, 1, 1, 0, 1, 2026, 212, 44, 13, 6, 4, 2, 1, 1, 0, 1, 3273, 325, 56, 21, 8, 6, 2, 2, 1, 1, 0, 1

Offset: 0

Alois P. Heinz, Mar 03 2013

For k > 0, column k is asymptotic to sqrt(3) * (2*k+1) * exp(Pi*sqrt(2*n/3)) / (2 * k^2 * (k+1)^2 * Pi^2) ~ 6 * (2*k+1) * n * p(n) / (k^2 * (k+1)^2 * Pi^2), where p(n) is the partition function A000041(n). - Vaclav Kotesovec, May 29 2018

			The partitions of n=4 are [1,1,1,1], [2,1,1], [2,2], [3,1], [4].  Parts <= 4 with multiplicity m=0 sum up to (2+3+4)+(3+4)+(1+3+4)+(2+4)+(1+2+3) = 36, for m=1 the sum is 2+(3+1)+4 = 10, for m=2 the sum is 1+2 = 3, for m=3 the sum is 0, for m=4 the sum is 1 => row 4 = [36, 10, 3, 0, 1].
Triangle T(n,k) begins:
    0;
    0,  1;
    3,  2,  1;
   11,  6,  0, 1;
   36, 10,  3, 0, 1;
   79, 21,  3, 1, 0, 1;
  186, 33,  7, 3, 1, 0, 1;
  345, 59,  9, 4, 1, 1, 0, 1;
  672, 89, 20, 4, 4, 1, 1, 0, 1;

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

Columns k=0-10 give: A213679, A103628, A117525, A222731, A222732, A222733, A222734, A222735, A222736, A222737, A222738.

Cf. A000041, A000217, A002865, A014153, A066186.

b:= proc(n, p) option remember; `if`(n=0 and p=0, [1, 0],
      `if`(p=0, [0$(n+2)], add((l-> subsop(m+2=p*l[1]+l[m+2], l))
          ([b(n-p*m, p-1)[], 0$(p*m)]), m=0..n/p)))
    end:
T:= n-> subsop(1=NULL, b(n, n))[]:
seq(T(n), n=0..14);

b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n - p*m, p-1] , Array[0&, p*m]]], {m, 0, n/p}]]]; Rest /@ Table[b[n, n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)

Sum_{k=0..n} k*T(n,k) = A066186(n) = n*A000041(n).
Sum_{k=1..n} T(n,k) = A014153(n-1) for n>0.
Sum_{k=0..n} T(n,k) = n*(n+1)/2*A000041(n) = A000217(n)*A000041(n).
(2 * Sum_{k=0..n} T(n,k)) / (Sum_{k=0..n} k*T(n,k)) = n+1 for n>0.
T(2*n+1,n+1) = A002865(n).

A325504 Product of products of parts over all strict integer partitions of n.

Original entry on oeis.org

1, 1, 2, 6, 12, 120, 1440, 40320, 1209600, 1567641600, 2633637888000, 13905608048640000, 5046067048690483200000, 5289893008483207348224000000, 1266933607446134946465526579200000000, 99304891373531545064656621572980736000000000000

Offset: 0

Gus Wiseman, May 07 2019

nonn

			The strict partitions of 5 are {(5), (4,1), (3,2)} with product a(5) = 5*4*1*3*2 = 120.
The sequence of terms together with their prime indices begins:
              1: {}
              1: {}
              2: {1}
              6: {1,2}
             12: {1,1,2}
            120: {1,1,1,2,3}
           1440: {1,1,1,1,1,2,2,3}
          40320: {1,1,1,1,1,1,1,2,2,3,4}
        1209600: {1,1,1,1,1,1,1,1,2,2,2,3,3,4}
     1567641600: {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4}
  2633637888000: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4}

Alois P. Heinz, Table of n, a(n) for n = 0..30

Cf. A000009, A006128, A007870 (non-strict version), A015723, A022629 (sum of products of parts), A066186, A066189, A066633, A246867, A325505, A325506, A325512, A325513, A325515.

a:= n-> mul(i, i=map(x-> x[], select(x->
        nops(x)=nops({x[]}), combinat[partition](n)))):
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 03 2021
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1$2], `if`(i<1, [0, 1], ((f, g)->
     [f[1]+g[1], f[2]*g[2]*i^g[1]])(b(n, i-1), b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 03 2021

Table[Times@@Join@@Select[IntegerPartitions[n],UnsameQ@@#&],{n,0,10}]

A001222(a(n)) = A325515(n).

a(n) = A003963(A325506(n)).

A206561 Triangle read by rows: T(n,k) = total sum of parts >= k in all partitions of n.

Original entry on oeis.org

1, 4, 2, 9, 5, 3, 20, 13, 7, 4, 35, 23, 15, 9, 5, 66, 47, 31, 19, 11, 6, 105, 75, 53, 35, 23, 13, 7, 176, 131, 93, 66, 42, 27, 15, 8, 270, 203, 151, 106, 74, 49, 31, 17, 9, 420, 323, 241, 178, 126, 86, 56, 35, 19, 10, 616, 477, 365, 272, 200, 140, 98, 63, 39, 21, 11

Offset: 1

Omar E. Pol, Feb 14 2012

From Omar E. Pol, Mar 18 2018: (Start)
In the n-th row of the triangle the first differences together with its last term give the n-th row of triangle A138785 (see below):
Row..........:  1     2       3            4               5          ...
               ---  ----   -------   ------------   ----------------
This triangle:  1;  4, 2;  9, 5, 3;  20, 13, 7, 4;  35, 23, 15, 9, 5; ...
                |   | /|   | /| /|    | / | /| /|    | / | / | /| /|
                |   |/ |   |/ |/ |    |/  |/ |/ |    |/  |/  |/ |/ |
A138785......:  1;  2, 2;  4, 2, 3;   7,  6, 3, 4;  12,  8,  6, 4, 5; ... (End)

			Triangle begins:
    1;
    4,  2;
    9,  5,  3;
   20, 13,  7,  4;
   35, 23, 15,  9,  5;
   66, 47, 31, 19, 11,  6;
  105, 75, 53, 35, 23, 13,  7;
  ...

Alois P. Heinz, Rows n = 1..141, flattened

Columns 1-2 give A066186, A194552.
Right border gives A000027.
Cf. A138785, A181187.
Row sums give A066183. - Omar E. Pol, Mar 19 2018
Both A180681 and A299768 have the same row sums as this triangle. - Omar E. Pol, Mar 21 2018

Table[With[{s = IntegerPartitions[n]}, Table[Total@ Flatten@ Map[Select[#, # >= k &] &, s], {k, n}]], {n, 11}] // Flatten (* Michael De Vlieger, Mar 19 2018 *)

T(n,n) = n, T(n,k) = T(n,k+1) + k * A066633(n,k) for k < n.

T(n,k) = Sum_{i=k..n} A138785(n,i).

More terms from Alois P. Heinz, Feb 14 2012

A220908 The second rank moment function N_2(n).

Original entry on oeis.org

0, 2, 8, 20, 42, 80, 140, 238, 380, 602, 910, 1372, 1996, 2900, 4102, 5790, 8002, 11046, 14980, 20282, 27090, 36092, 47546, 62510, 81374, 105700, 136210, 175084, 223510, 284694, 360410, 455244, 572054, 717160, 894964, 1114470, 1382032, 1710262, 2108750, 2594704, 3182120

Offset: 1

N. J. A. Sloane, Jan 02 2013

nonn

N_2(n) is also called the second Atkin-Garvan moment (see Andrews' paper). - Omar E. Pol, Oct 23 2013

Alois P. Heinz, Table of n, a(n) for n = 1..1000
G. E. Andrews, The number of smallest parts in the partitions of n, p. 3
F. G. Garvan, Higher-order spt functions, Adv. Math. 228 (2011), no. 1, 241-265. See Eq. (1.1).

Cf. A000041, A092269, A220907.

b:= proc(n, i) option remember; `if`(n=0 or i<1, 0,
      `if`(irem(n, i, 'r')=0, r, 0)+add(b(n-i*j, i-1), j=0..n/i))
    end:
a:= n-> 2*(n*combinat[numbpart](n)- b(n, n)):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 09 2013

terms = 41; gf = Sum[x^n/(1 - x^n)*Product[1/(1 - x^k), {k, n, terms}], {n, 1, terms}]; spt = CoefficientList[ Series[gf, {x, 0, terms}], x] // Rest; a[n_] := 2*(n*PartitionsP[n] - spt[[n]]); Table[a[n], {n, 1, terms}] (* Jean-François Alcover, Jan 17 2013, after g.f. of spt(n) *)

a(n) = 2*A220907(n) = 2*(n*A000041(n)-A092269(n)).
a(n) = 2*(A066186(n) - A092269(n)). - Omar E. Pol, Jan 09 2013
a(n) = A220909(n) - A211982(n). - Omar E. Pol, Jan 16 2013
a(n) ~ exp(Pi*sqrt(2*n/3))/(2*sqrt(3)) * (1 - (3*sqrt(6)/(2*Pi) + Pi/(24*sqrt(6)))/sqrt(n) + (5/48 + Pi^2/6912)/n). - Vaclav Kotesovec, Jul 31 2017

cp's OEIS Frontend

A340061 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(n-m) copies of m, with n >= 1 and m >= 1.

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A340793 Sequence whose partial sums give A000203.

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A343342 Number of integer partitions of n with no part dividing or divisible by all the others.

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A066967 Total sum of odd parts in all partitions of n.

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A220909 The second crank moment function M_2(n).

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A221649 Tetrahedron E(n,j,k) = k*T(j,k)*p(n-j), where T(j,k) = 1 if k divides j otherwise 0.

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A222730 Total sum T(n,k) of parts <= n of multiplicity k in all partitions of n; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A221649 Tetrahedron E(n,j,k) = kT(j,k)p(n-j), where T(j,k) = 1 if k divides j otherwise 0.