A002865 -id:A002865 - cp's OEIS frontend

A245095 Triangle read by rows: T(n,k) = A006218(k)*A002865(n-k).

1, 0, 3, 1, 0, 5, 1, 3, 0, 8, 2, 3, 5, 0, 10, 2, 6, 5, 8, 0, 14, 4, 6, 10, 8, 10, 0, 16, 4, 12, 10, 16, 10, 14, 0, 20, 7, 12, 20, 16, 20, 14, 16, 0, 23, 8, 21, 20, 32, 20, 28, 16, 20, 0, 27, 12, 24, 35, 32, 40, 28, 32, 20, 23, 0, 29, 14, 36, 40, 56, 40, 56, 32, 40, 23, 27, 0, 35

Offset: 1

Omar E. Pol, Jul 14 2014

Row sums give A006128, n >= 1.
Column 1 is A002865.
Leading diagonal is A006218, n >= 1.
For another version see A221530.

			Triangle begins:
  1;
  0,   3;
  1,   0,  5;
  1,   3,  0,  8;
  2,   3,  5,  0, 10;
  2,   6,  5,  8,  0, 14;
  4,   6, 10,  8, 10,  0, 16;
  4,  12, 10, 16, 10, 14,  0, 20;
  7,  12, 20, 16, 20, 14, 16,  0, 23;
  8,  21, 20, 32, 20, 28, 16, 20,  0, 27;
  12, 24, 35, 32, 40, 28, 32, 20, 23,  0, 29;
  14, 36, 40, 56, 40, 56, 32, 40, 23, 27,  0, 35;
  ...
For n = 6:
  -------------------------
  k   A006218        T(6,k)
  -------------------------
  1      1  *  2   =    2
  2      3  *  2   =    6
  3      5  *  1   =    5
  4      8  *  1   =    8
  5     10  *  0   =    0
  6     14  *  1   =   14
  .         A002865
  -------------------------
So row 6 is [2, 6, 5, 8, 0, 14] and the sum of row 6 is 2+6+5+8+0+14 = 35 equaling A006128(6) = 35.

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened)

Cf. A000005, A000041, A002865, A006128, A006218, A221529, A221530, A245099.

A245095row[n_]:=Accumulate[DivisorSigma[0,Range[n]]]Reverse[Differences[PartitionsP[Range[-1,n-1]]]];Array[A245095row,10] (* Paolo Xausa, Sep 04 2023 *)

a006218(n) = sum(k=1, n, n\k);
a002865(n) = if(n, numbpart(n)-numbpart(n-1), 1);
row(n) = vector(n, i, a006218(i)*a002865(n-i)); \\ Michel Marcus, Jul 18 2014

A182746 Bisection (even part) of number of partitions that do not contain 1 as a part A002865.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 21, 34, 55, 88, 137, 210, 320, 478, 708, 1039, 1507, 2167, 3094, 4378, 6153, 8591, 11914, 16424, 22519, 30701, 41646, 56224, 75547, 101066, 134647, 178651, 236131, 310962, 408046, 533623, 695578, 903811, 1170827, 1512301, 1947826, 2501928

Offset: 0

Omar E. Pol, Dec 01 2010

a(n+1) is the number of partitions p of 2n-1 such that (number of parts of p) is a part of p, for n >=0. - Clark Kimberling, Mar 02 2014

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Marco Baggio, Vasilis Niarchos, Kyriakos Papadodimas, and Gideon Vos, Large-N correlation functions in N = 2 superconformal QCD, arXiv preprint arXiv:1610.07612 [hep-th], 2016.
K. Blum, Bounds on the Number of Graphical Partitions, arXiv:2103.03196 [math.CO], 2021. See Table on p. 7.

Cf. A000041, A002865, A058696, A135010, A138121, A182740, A182742, A182743, A182747.

b:= proc(n, i) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i<2 then 0
    else b(n, i-1) +b(n-i, i)
      fi
    end:
a:= n-> b(2*n, 2*n):
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 01 2010

Table[Count[IntegerPartitions[2 n -1], p_ /; MemberQ[p, Length[p]]], {n, 20}]   (* Clark Kimberling, Mar 02 2014 *)
b[n_, i_] := b[n, i] = Which[n<0, 0, n==0, 1, i<2, 0, True, b[n, i-1] + b[n-i, i]]; a[n_] := b[2*n, 2*n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Sep 21 2015, after Alois P. Heinz *)
a[n_] := PartitionsP[2*n] - PartitionsP[2*n - 1]; Table[a[n], {n, 0, 40}] (* George Beck, Jun 05 2017 *)

a(n)=numbpart(2*n)-numbpart(2*n-1) \\ Charles R Greathouse IV, Jun 06 2017

a(n) = p(2*n) - p(2*n-1), where p is the partition function, A000041. - George Beck, Jun 05 2017 [Shifted by Georg Fischer, Jun 20 2022]

More terms from Alois P. Heinz, Dec 01 2010

A182747 Bisection (odd part) of number of partitions that do not contain 1 as a part A002865.

Original entry on oeis.org

0, 1, 2, 4, 8, 14, 24, 41, 66, 105, 165, 253, 383, 574, 847, 1238, 1794, 2573, 3660, 5170, 7245, 10087, 13959, 19196, 26252, 35717, 48342, 65121, 87331, 116600, 155038, 205343, 270928, 356169, 466610, 609237, 792906, 1028764, 1330772, 1716486, 2207851

Offset: 0

Omar E. Pol, Dec 01 2010

a(n+1) = number of partitions p of 2n such that (number of parts of p) is a part of p, for n >=0. - Clark Kimberling, Mar 02 2014

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A002865, A058695, A135010, A138121, A182740, A182742, A182743, A182746.

b:= proc(n,i) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i<2 then 0
    else b(n, i-1) +b(n-i, i)
      fi
    end:
a:= n-> b(2*n+1, 2*n+1):
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 01 2010

f[n_] := Table[PartitionsP[2 k + 1] - PartitionsP[2 k], {k, 0, n}] (* George Beck, Aug 14 2011 *)
(* also *)
Table[Count[IntegerPartitions[2 n], p_ /; MemberQ[p, Length[p]]], {n, 20}] (* Clark Kimberling, Mar 02 2014 *)
b[n_, i_] := b[n, i] = Which[n<0, 0, n == 0, 1, i<2, 0, True, b[n, i-1] + b[n-i, i]]; a[n_] := b[2*n+1, 2*n+1]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

a(n) = p(2*n+1)-p(2*n), where p is the partition function, A000041. - George Beck, Aug 14 2011

More terms from Alois P. Heinz, Dec 01 2010

A245099 Triangle read by rows: T(n,k) = A024916(k)*A002865(n-k).

Original entry on oeis.org

1, 0, 4, 1, 0, 8, 1, 4, 0, 15, 2, 4, 8, 0, 21, 2, 8, 8, 15, 0, 33, 4, 8, 16, 15, 21, 0, 41, 4, 16, 16, 30, 21, 33, 0, 56, 7, 16, 32, 30, 42, 33, 41, 0, 69, 8, 28, 32, 60, 42, 66, 41, 56, 0, 87, 12, 32, 56, 60, 84, 66, 82, 56, 69, 0, 99, 14, 48, 64

Offset: 1

Omar E. Pol, Jul 13 2014

Row sums give A066186.
Column 1 is A002865.
Leading diagonal is A024916.
Since A024916(k) has a symmetric representation then both T(n,k) and the partial sums of row n can be represented by symmetric polycubes - for more information see A237593 and A237270. For another version see A221529.

			Triangle begins:
1;
0,   4;
1,   0,  8;
1,   4,  0, 15;
2,   4,  8,  0, 21;
2,   8,  8, 15,  0, 33;
4,   8, 16, 15, 21,  0, 41;
4,  16, 16, 30, 21, 33,  0, 56;
7,  16, 32, 30, 42, 33, 41,  0, 69;
8,  28, 32, 60, 42, 66, 41, 56,  0, 87;
12, 32, 56, 60, 84, 66, 82, 56, 69,  0, 99;
...
For n = 6:
-------------------------
k   A024916        T(6,k)
-------------------------
1      1  *  2   =    2
2      4  *  2   =    8
3      8  *  1   =    8
4     15  *  1   =   15
5     21  *  0   =    0
6     33  *  1   =   33
.         A002865
-------------------------
So row 6 is [2, 8, 8, 15, 0, 33] and the sum of row 6 is 2+8+8+15+0+33 = 66 equaling A066186(6) = 6*A000041(6) = 6*11 = 66.

Cf. A000041, A000203, A002865, A024916, A066186, A221529, A221530, A245095.

A340426 Triangle read by rows: T(n,k) = A000203(n-k+1)*A002865(k-1), 1 <= k <= n.

Original entry on oeis.org

1, 3, 0, 4, 0, 1, 7, 0, 3, 1, 6, 0, 4, 3, 2, 12, 0, 7, 4, 6, 2, 8, 0, 6, 7, 8, 6, 4, 15, 0, 12, 6, 14, 8, 12, 4, 13, 0, 8, 12, 12, 14, 16, 12, 7, 18, 0, 15, 8, 24, 12, 28, 16, 21, 8, 12, 0, 13, 15, 16, 24, 14, 28, 28, 24, 12, 28, 0, 18, 13, 30, 16, 48, 24, 49, 32, 36, 14, 14, 0, 12

Offset: 1

Omar E. Pol, Jan 07 2021

Conjecture: the sum of row n equals A138879(n), the sum of all parts in the last section of the set of partitions of n.

			Triangle begins:
   1;
   3,  0;
   4,  0,  1;
   7,  0,  3,  1;
   6,  0,  4,  3,  2;
  12,  0,  7,  4,  6,  2;
   8,  0,  6,  7,  8,  6,  4;
  15,  0, 12,  6, 14,  8, 12,  4;
  13,  0,  8, 12, 12, 14, 16, 12,  7;
  18,  0, 15,  8, 24, 12, 28, 16, 21,  8;
  12,  0, 13, 15, 16, 24, 14, 28, 28, 24, 12;
  28,  0, 18, 13, 30, 16, 48, 24, 49, 32, 36, 14;
...
For n = 6 the calculation of every term of row 6 is as follows:
--------------------------
k   A002865         T(6,k)
--------------------------
1      1   *   12  =  12
2      0   *   6   =   0
3      1   *   7   =   7
4      1   *   4   =   4
5      2   *   3   =   6
6      2   *   1   =   2
.           A000203
--------------------------
The sum of row 6 is 12 + 0 + 7 + 4 + 6 + 2 = 31, equaling A138879(6) = 31.

Columns 1, 3 and 4 give A000203.
Column 2 gives A000004.
Columns 5 and 6 gives A074400.
Column 7 and 8 give A239050.
Column 9 gives A319527.
Column 10 gives A319528.
Leading diagonal gives A002865.
Cf. A135010, A138879.

A340424 Triangle read by rows: T(n,k) = A024916(n-k+1)*A002865(k-1), 1 <= k <= n.

Original entry on oeis.org

1, 4, 0, 8, 0, 1, 15, 0, 4, 1, 21, 0, 8, 4, 2, 33, 0, 15, 8, 8, 2, 41, 0, 21, 15, 16, 8, 4, 56, 0, 33, 21, 30, 16, 16, 4, 69, 0, 41, 33, 42, 30, 32, 16, 7, 87, 0, 56, 41, 66, 42, 60, 32, 28, 8, 99, 0, 69, 56, 82, 66, 84, 60, 56, 32, 12, 127, 0, 87, 69, 112, 82, 132, 84, 105, 64, 48, 14

Offset: 1

Omar E. Pol, Jan 07 2021

Conjecture: the sum of row n equals A066186(n), the sum of all parts of all partitions of n.

			Triangle begins:
   1;
   4,  0;
   8,  0,  1;
  15,  0,  4,  1;
  21,  0,  8,  4,  2;
  33,  0, 15,  8,  8,  2;
  41,  0, 21, 15, 16   8,  4;
  56,  0, 33, 21, 30, 16, 16,  4;
  69,  0, 41, 33, 42, 30, 32, 16,  7;
  87,  0, 56, 41, 66, 42, 60, 32, 28,  8;
  99,  0, 69, 56, 82, 66, 84, 60, 56, 32, 12;
...
For n = 6 the calculation of every term of row 6 is as follows:
--------------------------
k   A002865         T(6,k)
--------------------------
1      1   *  33   =  33
2      0   *  21   =   0
3      1   *  15   =  15
4      1   *   8   =   8
5      2   *   4   =   8
6      2   *   1   =   2
.           A024916
--------------------------
The sum of row 6 is 33 + 0 + 15 + 8 + 8 + 2 = 66, equaling A066186(6) = 66.

Mirror of A245099.
Columns 1, 3 and 4 are A024916 (partial sums of A000203).
Column 2 gives A000004.
Columns 5 and 6 give A327329.
Columns 7 and 8 give A243980.
Leading diagonal gives A002865.
Cf. A066186.

A122402 A sequence of rectangular tables with row sums A122401 related to A002865 and A122172.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 3, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 3, 6, 7, 6, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 3, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 3, 6, 8, 8, 6, 3, 1

Offset: 0

Alford Arnold, Sep 02 2006

Sequence A002865 gives the number of rows in each rectangular table. A122402 is a subsequence of A122172.

			There are four cyclic partitions of 6: 6, 42, 33 and 222; therefore the corresponding table is
1 1 1 1 1 1 1
1 2 3 3 3 2 1
1 2 3 4 3 2 1
1 3 6 7 6 3 1
The tables begin:
1
0
1 1 1
1 1 1 1
1 1 1 1 1
1 2 3 2 1
1 1 1 1 1 1
1 2 3 3 2 1
...

Cf. A002865 A122172 A122401.

A340524 Triangle read by rows: T(n,k) = A000005(n-k+1)*A002865(k-1), 1 <= k <= n.

Original entry on oeis.org

1, 2, 0, 2, 0, 1, 3, 0, 2, 1, 2, 0, 2, 2, 2, 4, 0, 3, 2, 4, 2, 2, 0, 2, 3, 4, 4, 4, 4, 0, 4, 2, 6, 4, 8, 4, 3, 0, 2, 4, 4, 6, 8, 8, 7, 4, 0, 4, 2, 8, 4, 12, 8, 14, 8, 2, 0, 3, 4, 4, 8, 8, 12, 14, 16, 12, 6, 0, 4, 3, 8, 4, 16, 8, 21, 16, 24, 14, 2, 0, 2, 4, 6, 8, 8, 16, 14, 24, 24, 28, 21

Offset: 1

Omar E. Pol, Jan 10 2021

Conjecture: the sum of row n equals A138137(n), the total number of parts in the last section of the set of partitions of n.

			Triangle begins:
1;
2, 0;
2, 0, 1;
3, 0, 2, 1;
2, 0, 2, 2, 2;
4, 0, 3, 2, 4, 2;
2, 0, 2, 3, 4, 4,  4;
4, 0, 4, 2, 6, 4,  8,  4;
3, 0, 2, 4, 4, 6,  8,  8,  7;
4, 0, 4, 2, 8, 4, 12,  8, 14,  8;
2, 0, 3, 4, 4, 8,  8, 12, 14, 16, 12;
6, 0, 4, 3, 8, 4, 16,  8, 21, 16, 24, 14;
2, 0, 2, 4, 6, 8,  8, 16, 14, 24, 24, 28, 21;
...
For n = 6 the calculation of every term of row 6 is as follows:
--------------------------
k   A002865         T(6,k)
--------------------------
1      1   *   4   =   4
2      0   *   2   =   0
3      1   *   3   =   3
4      1   *   2   =   2
5      2   *   2   =   4
6      2   *   1   =   2
.           A000005
--------------------------
The sum of row 6 is 4 + 0 + 3 + 2 + 4 + 2 = 15, equaling A138137(6) = 15.

Row sums give A138137 (conjectured).
Columns 1, 3 and 4 are A000005.
Column 2 gives A000004.
Columns 5 and 6 give A062011.
Columns 7 and 8 give A145154, n >= 1.
Leading diagonal gives A002865.
Cf. A339304 (irregular or expanded version).
Cf. A135010, A138121, A221531, A336811, A339106, A340424, A340426.

f(n) = if (n==0, 1, numbpart(n) - numbpart(n-1)); \\ A002865
T(n, k) = numdiv(n-k+1) * f(k-1); \\ Michel Marcus, Jan 13 2021

A340583 Triangle read by rows: T(n,k) = A002865(n-k)*A000203(k), 1 <= k <= n.

Original entry on oeis.org

1, 0, 3, 1, 0, 4, 1, 3, 0, 7, 2, 3, 4, 0, 6, 2, 6, 4, 7, 0, 12, 4, 6, 8, 7, 6, 0, 8, 4, 12, 8, 14, 6, 12, 0, 15, 7, 12, 16, 14, 12, 12, 8, 0, 13, 8, 21, 16, 28, 12, 24, 8, 15, 0, 18, 12, 24, 28, 28, 24, 24, 16, 15, 13, 0, 12, 14, 36, 32, 49, 24, 48, 16, 30, 13, 18, 0, 28

Offset: 1

Omar E. Pol, Jan 15 2021

T(n,k) is the total number of cubic cells added at n-th stage to the right prisms whose bases are the parts of the symmetric representation of sigma(k) in the polycube described in A221529.

Partial sums of column k gives the column k of A221529.

			Triangle begins:
   1;
   0,  3;
   1,  0,  4;
   1,  3,  0,  7;
   2,  3,  4,  0,  6;
   2,  6,  4,  7,  0, 12;
   4,  6,  8,  7,  6,  0,  8;
   4, 12,  8, 14,  6, 12,  0, 15;
   7, 12, 16, 14, 12, 12,  8,  0, 13;
   8, 21, 16, 28, 12, 24,  8, 15,  0, 18;
  12, 24, 28, 28, 24, 24, 16, 15, 13,  0, 12;
  14, 36, 32, 49, 24, 48, 16, 30, 13, 18,  0, 28;
...
For n = 6 the calculation of every term of row 6 is as follows:
--------------------------
k   A000203         T(6,k)
--------------------------
1      1   *   2  =    2
2      3   *   2   =   6
3      4   *   1   =   4
4      7   *   1   =   7
5      6   *   0   =   0
6     12   *   1   =  12
.           A002865
--------------------------
The sum of row 6 is 2 + 6 + 4 + 7 + 0 + 12 = 31, equaling A138879(6).

Row sums give A138879.
Column 1 gives A002865.
Diagonals 1, 3 and 4 give A000203.
Diagonal 2 gives A000004.
Diagonals 5 and 6 give A074400.
Diagonals 7 and 8 give A239050.
Diagonal 9 gives A319527.
Diagonal 10 gives A319528.
Cf. A221529 (partial column sums).
Cf. A340426 (mirror).
Cf. A135010, A221530, A245095, A245099, A339278.

A340583[n_, k_] := (PartitionsP[n - k] - PartitionsP[(n - k) - 1])*
   DivisorSigma[1, k];
Table[A340583[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Robert P. P. McKone, Jan 25 2021 *)

A182844 a(0)=0: a(n)=A002865(2n)+A002865(2n+1), n>=1.

Original entry on oeis.org

0, 2, 4, 8, 15, 26, 45, 75, 121, 193, 302, 463, 703, 1052, 1555, 2277, 3301, 4740, 6754, 9548, 13398, 18678, 25873, 35620, 48771, 66418, 89988, 121345, 162878, 217666, 289685, 383994, 507059, 667131, 874656, 1142860, 1488484, 1932575, 2501599, 3228787

Offset: 0

Omar E. Pol, Jan 24 2011

a(n) is also the length of the n-th "large mirror" of the "mirror" version of the shell model of partitions A135010.

			a(0)=0 by definition. a(1)=1+1=2: a(2)=2+2=4. a(3)=4+4=8. a(4)=7+8=15. a(5)=12+14=26. a(6)=21+24=45.

Nathaniel Johnston, Table of n, a(n) for n = 0..1000

Cf. A002865, A135010. For another version see A182845.

Extended by Nathaniel Johnston, May 06 2011

cp's OEIS Frontend

A245095 Triangle read by rows: T(n,k) = A006218(k)*A002865(n-k).

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A182746 Bisection (even part) of number of partitions that do not contain 1 as a part A002865.

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A182747 Bisection (odd part) of number of partitions that do not contain 1 as a part A002865.

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A245099 Triangle read by rows: T(n,k) = A024916(k)*A002865(n-k).

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A340426 Triangle read by rows: T(n,k) = A000203(n-k+1)*A002865(k-1), 1 <= k <= n.

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A340424 Triangle read by rows: T(n,k) = A024916(n-k+1)*A002865(k-1), 1 <= k <= n.

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A122402 A sequence of rectangular tables with row sums A122401 related to A002865 and A122172.

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A340524 Triangle read by rows: T(n,k) = A000005(n-k+1)*A002865(k-1), 1 <= k <= n.

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A340583 Triangle read by rows: T(n,k) = A002865(n-k)*A000203(k), 1 <= k <= n.

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A182844 a(0)=0: a(n)=A002865(2n)+A002865(2n+1), n>=1.