A182701 -id:A182701 - cp's OEIS frontend

A182705 Row sums of triangle A182701.

1, 4, 12, 28, 60, 114, 210, 360, 603, 970, 1529, 2340, 3536, 5222, 7620, 10944, 15555, 21816, 30343, 41740, 56994, 77132, 103684, 138312, 183450, 241696, 316764, 412776, 535340, 690750, 887499, 1135072, 1446060, 1834742, 2319555, 2921616, 3667921, 4589260

Offset: 1

Omar E. Pol, Nov 28 2010

nonn

Robert Price, Table of n, a(n) for n = 1..2000

Cf. A182700, A182701, A182702, A182704.

Total /@ Table[n*PartitionsP[n-k], {n, 38}, {k, n}] // Flatten (* Robert Price, Jun 23 2020 *)

a000070(n) = sum(k=0, n, numbpart(k));
for(n=1, 100, print1(n*a000070(n - 1), ", ")) \\ Indranil Ghosh, Jun 08 2017

from sympy import npartitions as p
def a000070(n): return sum([p(k) for k in range(n + 1)])
def a(n): return n*a000070(n - 1) # Indranil Ghosh, Jun 08 2017

a(n) = n * A000070(n-1).

G.f.: x*f'(x), where f(x) = (x/(1 - x))*Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Jun 08 2017

A182700 Triangle T(n,k) = n*A000041(n-k), 0<=k<=n, read by rows.

Original entry on oeis.org

0, 1, 1, 4, 2, 2, 9, 6, 3, 3, 20, 12, 8, 4, 4, 35, 25, 15, 10, 5, 5, 66, 42, 30, 18, 12, 6, 6, 105, 77, 49, 35, 21, 14, 7, 7, 176, 120, 88, 56, 40, 24, 16, 8, 8, 270, 198, 135, 99, 63, 45, 27, 18, 9, 9, 420, 300, 220, 150, 110, 70, 50, 30, 20, 10, 10, 616, 462, 330, 242, 165, 121, 77, 55, 33

Offset: 0

Omar E. Pol, Nov 27 2010

T(n,k) is the sum of the parts of all partitions of n that contain k as a part, assuming that all partitions of n have 0 as a part: Thus, column 0 gives the sum of the parts of all partitions of n.
By definition all entries in row n>0 are divisible by n.
Row sums are 0, 2, 8, 21, 48, 95, 180, 315, 536, 873, 1390, 2145,...
The partitions of n+k that contain k as a part can be obtained by adding k to every partition of n assuming that all partitions of n have 0 as a part.
For example, the partitions of 6+k that contain k as a part are
k + 6
k + 3 + 3
k + 4 + 2
k + 2 + 2 + 2
k + 5 + 1
k + 3 + 2 + 1
k + 4 + 1 + 1
k + 2 + 2 + 1 + 1
k + 3 + 1 + 1 + 1
k + 2 + 1 + 1 + 1 + 1
k + 1 + 1 + 1 + 1 + 1 + 1
The partition number A000041(n) is also the number of partitions of m*(n+k) into parts divisible by m and that contain m*k as a part, with k>=0, m>=1, n>=0 and assuming that all partitions of n have 0 as a part.

			For n=7 and k=4 there are 3 partitions of 7 that contain 4 as a part. These partitions are (4+3)=7, (4+2+1)=7 and (4+1+1+1)=7. The sum is 7+7+7 = 7*3 = 21. By other way, the partition number of 7-4 is A000041(3) = p(3)=3, then 7*3 = 21, so T(7,4) = 21.
Triangle begins with row n=0 and columns 0<=k<=n :
0,
1, 1,
4, 2, 2,
9, 6, 3, 3,
20,12,8, 4, 4,
35,25,15,10,5, 5,
66,42,30,18,12,6, 6

Robert Price, Table of n, a(n) for n = 0..5150 (First 100 rows)

Cf. A000041, A027293, A135010, A138121.
Two triangles that are essentially the same as this are A027293 and A140207. - N. J. A. Sloane, Nov 28 2010
Row sums give A182704.

A182700 := proc(n,k) n*combinat[numbpart](n-k) ; end proc:
seq(seq(A182700(n,k),k=0..n),n=0..15) ;

Table[n*PartitionsP[n-k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert Price, Jun 23 2020 *)

PARI
```
A182700(n,k) = n*numbpart(n-k)
```

T(n,0) = A066186(n).
T(n,k) = A182701(n,k), n>=1 and k>=1.
T(n,n) = n = min { T(n,k); 0<=k<=n }.

A182702 Triangle T(n,k) = n*(A000041(n-k)) read by rows, k>=0.

Original entry on oeis.org

1, 4, 2, 9, 6, 3, 20, 12, 8, 4, 35, 25, 15, 10, 5, 66, 42, 30, 18, 12, 6, 105, 77, 49, 35, 21, 14, 7, 176, 120, 88, 56, 40, 24, 16, 8, 270, 198, 135, 99, 63, 45, 27, 18, 9, 420, 300, 220, 150, 110, 70, 50, 30, 20, 10, 616, 462, 330, 242, 165, 121, 77, 55, 33, 22, 11

Offset: 1

Omar E. Pol, Nov 28 2010

The same as A182700, but without the last term of row n.

			Triangle begins:
1;
4, 2;
9, 6, 3;
20, 12, 8, 4;
35, 25, 15, 10, 5;
66, 42, 30, 18, 12, 6;

Robert Price, Table of n, a(n) for n = 1..5050 (First 100 rows)

Cf. A000041, A135010, A138121, A182701, A182702.

Column 1 give A066186.

Table[n*PartitionsP[n-k], {n, 0, 11}, {k, 0,  n - 1}] // Flatten (* Robert Price, Jun 23 2020 *)

tabl(nn) = {for (n = 1, nn, for (k = 0, n-1, print1(n*numbpart(n-k), ", ");); print(););} \\ Michel Marcus, Feb 13 2014

More terms from Michel Marcus, Feb 13 2014

A182704 Row sums of triangle A182700.

Original entry on oeis.org

0, 2, 8, 21, 48, 95, 180, 315, 536, 873, 1390, 2145, 3264, 4849, 7112, 10260, 14640, 20604, 28746, 39653, 54280, 73626, 99176, 132549, 176112, 232400, 305032, 398034, 516880, 667725, 858870

Offset: 0

Omar E. Pol, Nov 28 2010

nonn

Robert Price, Table of n, a(n) for n = 0..2000 (First 2000 rows)

Cf. A182700, A182701, A182705.

Total /@ Table[n*PartitionsP[n-k], {n, 0, 30}, {k, 0, n}]  (* Robert Price, Jun 23 2020 *)

a(n) = n*A000070(n).

G.f.: x*f'(x), where f(x) = (1/(1 - x))*Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Apr 13 2017

A182706 Row sums of triangle A182702.

Original entry on oeis.org

1, 6, 18, 44, 90, 174, 308, 528, 864, 1380, 2134, 3252, 4836, 7098, 10245, 14624, 20587, 28728, 39634, 54260, 73605, 99154, 132526, 176088, 232375, 305006, 398007, 516852, 667696, 858840

Offset: 1

Omar E. Pol, Nov 28 2010

nonn

Robert Price, Table of n, a(n) for n = 1..2000 (First 2000 rows)

Cf. A026905, A182700, A182701, A182702, A182703, A182704, A182705.

Total /@ Table[n*PartitionsP[n-k], {n, 30}, {k, 0, n - 1}] // Flatten (* Robert Price, Jun 23 2020 *)

a(n) = n * A026905(n).

A228816 Sum of all parts of all partitions of n that contain 1 as a part.

Original entry on oeis.org

1, 2, 6, 12, 25, 42, 77, 120, 198, 300, 462, 672, 1001, 1414, 2025, 2816, 3927, 5346, 7315, 9800, 13167, 17424, 23046, 30120, 39375, 50908, 65772, 84280, 107822, 136950, 173724, 218944, 275517, 344862, 430850, 535788, 665149, 822206, 1014585, 1247400

Offset: 1

Omar E. Pol, Sep 23 2013

nonn

Column 1 of A182701.

Cf. A000041, A000070, A066186, A138880, A194552, A228823.

Table[Total[Flatten[Select[IntegerPartitions[n],MemberQ[#,1]&]]],{n,40}] (* Harvey P. Dale, Sep 27 2015 *)

a(n) = n*A000041(n-1).
a(n) = A066186(n-1) + A000041(n-1), n >= 2.
a(n) = A194552(n-1) + A000070(n-1), n >= 2.

A228823 Triangle read by rows: T(n,k) = total number of parts in all partitions of n that contain k as a part, n>=1, 1<=k<=n.

Original entry on oeis.org

1, 2, 1, 5, 2, 1, 9, 5, 2, 1, 17, 9, 5, 2, 1, 27, 17, 9, 5, 2, 1, 46, 27, 17, 9, 5, 2, 1, 69, 46, 27, 17, 9, 5, 2, 1, 108, 69, 46, 27, 17, 9, 5, 2, 1, 158, 108, 69, 46, 27, 17, 9, 5, 2, 1, 234, 158, 108, 69, 46, 27, 17, 9, 5, 2, 1, 331, 234, 158, 108, 69

Offset: 1

Omar E. Pol, Sep 25 2013

Row n lists the first n elements of A093694 in decreasing order.

			Triangle begins:
1;
2,     1;
5,     2,   1;
9,     5,   2,   1;
17,    9,   5,   2,  1;
27,   17,   9,   5,  2,  1;
46,   27,  17,   9,  5,  2,  1;
69,   46,  27,  17,  9,  5,  2,  1;
108,  69,  46,  27, 17,  9,  5,  2,  1;
158, 108,  69,  46, 27, 17,  9,  5,  2,  1;
234, 158, 108,  69, 46, 27, 17,  9,  5,  2,  1;
331, 234, 158, 108, 69, 46, 27, 17,  9,  5,  2,  1;

Cf. A000041, A006128, A027293, A093694, A182701.

T(n,k) = A000041(n-k) + A006128(n-k) = A093694(n-k).

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A182705 Row sums of triangle A182701.

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A182700 Triangle T(n,k) = n*A000041(n-k), 0<=k<=n, read by rows.

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A182702 Triangle T(n,k) = n*(A000041(n-k)) read by rows, k>=0.

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A182704 Row sums of triangle A182700.

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A182706 Row sums of triangle A182702.

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A228816 Sum of all parts of all partitions of n that contain 1 as a part.

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A228823 Triangle read by rows: T(n,k) = total number of parts in all partitions of n that contain k as a part, n>=1, 1<=k<=n.

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