A026820 -id:A026820 - cp's OEIS frontend

A058400 Triangle of partial row sums of partition triangle A058398.

1, 2, 1, 3, 2, 1, 5, 4, 3, 1, 7, 6, 5, 3, 1, 11, 10, 9, 7, 4, 1, 15, 14, 13, 11, 8, 4, 1, 22, 21, 20, 18, 15, 10, 5, 1, 30, 29, 28, 26, 23, 18, 12, 5, 1, 42, 41, 40, 38, 35, 30, 23, 14, 6, 1, 56, 55, 54, 52, 49, 44, 37, 27, 16, 6, 1, 77, 76, 75, 73, 70, 65, 58, 47, 34, 19, 7, 1, 101

Offset: 1

Wolfdieter Lang, Dec 11 2000

Mirror of A026820. - Omar E. Pol, Apr 21 2012

			Triangle begins:
1;
2,   1;
3,   2,  1;
5,   4,  3,  1;
7,   6,  5,  3, 1;
11, 10,  9,  7, 4, 1;
15, 14, 13, 11, 8, 4, 1;

Columns 1-3: A000041(n), A000065(n), A007042(n+1).

Cf. A008284.

a(n, m) = sum(A058398(n, k), k=m..n).

A171985 Number of partitions of 2*n-1 into parts not greater than n.

Original entry on oeis.org

1, 2, 5, 11, 23, 44, 82, 146, 252, 423, 695, 1116, 1763, 2738, 4192, 6334, 9459, 13968, 20425, 29588, 42496, 60547, 85628, 120246, 167762, 232605, 320635, 439544, 599412, 813360, 1098480, 1476870, 1977087, 2635869, 3500382, 4630932, 6104533, 8019131, 10499093

Offset: 1

Reinhard Zumkeller, Jan 21 2010

nonn

Central terms of the triangle in A026820: a(n)=A026820(2*n-1,n).

			a(4) = (partitions of 7 into parts <= 4) = #{4+3, 4+2+1, 4+1+1+1, 3+3+1, 3+2+2, 3+2+1+1, 3+1+1+1+1, 2+2+2+1, 2+2+1+1+1, 2+1+1+1+1+1, 1+1+1+1+1+1+1} = 11.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

g:= proc(n, i) option remember;
     `if`(n=0, 1, `if`(i>1, g(n, i-1), 0)+`if`(i>n, 0, g(n-i, i)))
    end:
a:= n-> g(2*n-1, n):
seq (a(n), n=1..40);  # Alois P. Heinz, Jul 18 2012

g[n_, i_] := g[n, i] = If[n==0, 1, If[i>1, g[n, i-1], 0]+If[i>n, 0, g[n-i, i]]]; a[n_] := g[2*n-1, n]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 16 2017, after Alois P. Heinz *)

A233322 Triangle read by rows: T(n,k) = number of palindromic partitions of n in which no part exceeds k, 1 <= k <= n.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 3, 3, 4, 1, 2, 3, 3, 4, 1, 4, 5, 6, 6, 7, 1, 2, 5, 5, 6, 6, 7, 1, 5, 7, 10, 10, 11, 11, 12, 1, 3, 7, 8, 10, 10, 11, 11, 12, 1, 6, 9, 14, 15, 17, 17, 18, 18, 19, 1, 3, 9, 11, 15, 15, 17, 17, 18, 18, 19, 1, 7, 12, 20, 22, 26, 26, 28, 28, 29, 29, 30

Offset: 1

L. Edson Jeffery, Dec 10 2013

See A025065 for a definition of palindromic partition.

			Triangle begins:
1;
1, 2;
1, 1,  2;
1, 3,  3,  4;
1, 2,  3,  3,  4;
1, 4,  5,  6,  6,  7;
1, 2,  5,  5,  6,  6,  7;
1, 5,  7, 10, 10, 11, 11, 12;
1, 3,  7,  8, 10, 10, 11, 11, 12;
1, 6,  9, 14, 15, 17, 17, 18, 18, 19;
1, 3,  9, 11, 15, 15, 17, 17, 18, 18, 19;
1, 7, 12, 20, 22, 26, 26, 28, 28, 29, 29, 30;
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Cf. A025065, A026820; partial sums of row entries of A233321.

Cf. A233323, A233324 (palindromic compositions of n).

(* run this first: *)
Needs["Combinatorica`"];
(* run the following in a different cell: *)
a233321[n_] := {}; ; Do[Do[a = Partitions[n]; count = 0; Do[If[Max[a[[j]]] == k, x = Permutations[a[[j]]]; Do[If[x[[m]] == Reverse[x[[m]]], count++; Break[]], {m, Length[x]}]], {j, Length[a]}]; AppendTo[a233321[n], count], {k, n}], {n, nmax}]; a233322[n_] := {}; Do[Do[AppendTo[a233322[n], Sum[a233321[n][[j]], {j, k}]], {k, n}], {n,nmax}]; Table[a233322[n], {n, nmax}](* L. Edson Jeffery, Oct 09 2017 *)

\\ here PartitionCount is A026820.
PartitionCount(n,maxpartsize)={my(t=0); forpart(p=n, t++, maxpartsize); t}
T(n,k)=sum(i=0, (k-n%2)\2, PartitionCount(n\2-i, k));
for(n=1, 10, for(k=1, n, print1(T(n,k), ", ")); print) \\ Andrew Howroyd, Oct 09 2017

T(n,k) = Sum_{i=1..k} A233321(n,i).

T(n,k) = Sum_{i=0..(k+2*floor(n/2)-n)/2} A026820(floor(n/2)-i, k). - Andrew Howroyd, Oct 09 2017

Corrected row 7 as communicated by Andrew Howroyd. - L. Edson Jeffery, Oct 09 2017

A382041 Triangle read by rows: T(n, k) is the number of partitions of n with at most k parts where 0 <= k <= n, and each part is one of four kinds.

Original entry on oeis.org

1, 0, 4, 0, 4, 14, 0, 4, 20, 40, 0, 4, 30, 70, 105, 0, 4, 36, 116, 196, 252, 0, 4, 46, 170, 350, 490, 574, 0, 4, 52, 236, 556, 896, 1120, 1240, 0, 4, 62, 310, 845, 1505, 2079, 2415, 2580, 0, 4, 68, 400, 1200, 2400, 3584, 4480, 4960, 5180, 0, 4, 78, 494, 1670, 3626, 5910, 7842, 9162, 9822, 10108

Offset: 0

Peter Dolland, Mar 12 2025

Two unrestricted unary predicates on the parts set result in four kinds: The intersection, the both differences and the complement of the union.
The 1-kind case is Euler's table A026820.
The 2-kind case is A381895.
The 3-kind case is A382025.

			Triangle starts:
  0 : [1]
  1 : [0, 4]
  2 : [0, 4, 14]
  3 : [0, 4, 20,  40]
  4 : [0, 4, 30,  70,  105]
  5 : [0, 4, 36, 116,  196,  252]
  6 : [0, 4, 46, 170,  350,  490,  574]
  7 : [0, 4, 52, 236,  556,  896, 1120, 1240]
  8 : [0, 4, 62, 310,  845, 1505, 2079, 2415, 2580]
  9 : [0, 4, 68, 400, 1200, 2400, 3584, 4480, 4960, 5180]
 10 : [0, 4, 78, 494, 1670, 3626, 5910, 7842, 9162, 9822, 10108]
 ...

Main diagonal gives A023003.

Cf. A026820, A381895, A382025.

from sympy import binomial
from sympy.utilities.iterables import partitions
from sympy.combinatorics.partitions import IntegerPartition
kinds = 4 - 1   # the number of part kinds - 1
def a382041_row( n):
    if n == 0 : return [1]
    t = list( [0] * n)
    for p in partitions( n):
        p = IntegerPartition( p).as_dict()
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= binomial( kinds + p[k], kinds)
        if s > 0 :
            t[s - 1] += fact
    for i in range( n - 1):
        t[i+1] += t[i]
    return [0] + t

A383352 Triangle read by rows: T(n, k) is the number of partitions of a 2-colored set of n objects into at most k parts where 0 <= k <= n, and each part is one of 2 kinds.

Original entry on oeis.org

1, 0, 4, 0, 6, 16, 0, 8, 32, 52, 0, 10, 63, 123, 158, 0, 12, 100, 264, 384, 440, 0, 14, 158, 506, 876, 1086, 1170, 0, 16, 224, 896, 1800, 2500, 2836, 2956, 0, 18, 317, 1491, 3489, 5359, 6542, 7046, 7211, 0, 20, 420, 2372, 6324, 10848, 14208, 16056, 16776, 16996

Offset: 0

Peter Dolland, Apr 24 2025

			Triangle starts:
 0 : [1]
 1 : [0,  4]
 2 : [0,  6,  16]
 3 : [0,  8,  32,   52]
 4 : [0, 10,  63,  123,   158]
 5 : [0, 12, 100,  264,   384,   440]
 6 : [0, 14, 158,  506,   876,  1086,  1170]
 7 : [0, 16, 224,  896,  1800,  2500,  2836,  2956]
 8 : [0, 18, 317, 1491,  3489,  5359,  6542,  7046,  7211]
 9 : [0, 20, 420, 2372,  6324, 10848, 14208, 16056, 16776, 16996]
10 : [0, 22, 556, 3608, 11002, 20836, 29488, 34976, 37700, 38690, 38976]
...

Row sums of A383351.

Cf. A381895 (1-color), A381891 (1-kind), A026820 (1-color, 1-kind).

from sympy import binomial
from sympy.utilities.iterables import partitions
def calc_w(k , m):
    s = 0
    for p in partitions(m, m=k+1):
        fact = 1
        j = k + 1
        for x in p :
            fact *= binomial(j, p[x]) * (x + 1) ** p[x]
            j -= p[x]
        s += fact
    return s
def t_row(n):
    if n == 0 : return [1]
    t = list([0] * n)
    for p in partitions( n):
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= calc_w(k, p[k])
        if s > 0 :
            t[s - 1] += fact
    for i in range(n - 1):
        t[i + 1] += t[i]
    return [0] + t

T(n,k) = Sum_{i=0..k} A383351(n,i).

T(n,1) = 2*n + 2 for n >= 1.

A106254 Partition table in square format.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 3, 2, 1, 1, 3, 4, 3, 2, 1, 1, 4, 5, 5, 3, 2, 1, 1, 4, 7, 6, 5, 3, 2, 1, 1, 5, 8, 9, 7, 5, 3, 2, 1, 1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1, 7, 16, 23, 23, 20, 15, 11, 7, 5, 3, 2, 1

Offset: 1

Gary W. Adamson, Apr 27 2005

The square is the following table:
1 1 1 1 1 1...
2 2 2 2 2 2...
2 3 3 3 3 3...
3 4 5 5 5 5...
3 5 6 7 7 7...
4 7 9 10 11 11...
4 8 11 13 14 15...

			The partitions of 6 are:
1 + 1 + 1 + 1 + 1 + 1; 2 + 1 + 1 + 1 + 1; 3 + 1 + 1 + 1; 2 + 2 + 1 + 1; 4 + 1 + 1; 3 + 2 + 1; 2 + 2 + 2; 5 + 1, 4 + 2, 3 + 3, 6.
There are 9 partitions of 6 having summands no larger than 4, so p_4(6) = 9.

Ivan Niven, "Mathematics of Choice, How to Count Without Counting", MAA, 1965, pp. 98-99 (table p. 98).

Martin Y. Champel, Table of n, a(n) for n = 1..405
A. Comtet, S. N. Majumdar and S. Ouvry, Integer partitions and exclusion statistics, J. Phys. A: Math. Theor. vol. 40 (2007) pp. 11255-11269.
A. Comtet et al., Integer Partitions and Exclusion Statistics, arXiv:0705.2640 [cond-mat.stat-mech], 2007, eq. (12).

Essentially the same as A008284, except for missing one diagonal thereof (which would be zero row of this array).

T[n_, k_] := T[n, k] = If[n == 0 || k == 1, 1, T[n, k-1] + If[k > n, 0, T[n-k, k]]];
Table[T[n-k+1, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 08 2018, after Alois P. Heinz *)

Antidiagonals of table of values of p_k(n) (the number of partitions of n with no summand greater than k).

T(n,m) = sum_{i=1..m} A008284(n,i). T(n,m) = A026820(n,m) if m<=n and T(n,m)=T(n,n) if m>=n. G.f. column m: 1/(1-x)/(1-x^2)/.../(1-x^m) = sum_(n=1,2,3..) T(n,m) x^n [Comtet]. - R. J. Mathar, Aug 31 2007

Edited, corrected and extended by Franklin T. Adams-Watters, Jan 12 2006

A318806 Triangular array read by rows, where T(n,k) is the number of almost distinct partitions of n in which every part is <= k for 1 <= k <= n.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 4, 5, 6, 1, 2, 4, 6, 7, 8, 1, 2, 4, 7, 9, 10, 11, 1, 2, 4, 7, 10, 12, 13, 14, 1, 2, 4, 8, 12, 15, 17, 18, 19, 1, 2, 4, 8, 13, 17, 20, 22, 23, 24, 1, 2, 4, 8, 14, 20, 24, 27, 29, 30, 31, 1, 2, 4, 8, 15, 22, 28, 32, 35, 37, 38, 39, 1, 2, 4, 8, 15, 24, 32, 38, 42, 45, 47

Offset: 1

Sara Billey, Sep 04 2018

An almost distinct partition of n with parts bounded by k is a decreasing sequence of positive integers (a(1), a(2), ..., a(k)) such that n = a(1) + a(2) +...+ a(k), any a(i) > 1 is distinct from all other values, and all a(i) <= k.

			There are T(5,6) = 7 almost distinct partitions of 6 in which every part is <= 5: [5,1], [4,2], [4,1,1], [3,2,1], [3,1,1,1], [2,1,1,1,1], [1,1,1,1,1,1].
Triangle starts:
1;
1, 2;
1, 2, 3;
1, 2, 3, 4;
1, 2, 4, 5,  6;
1, 2, 4, 6,  7,  8;
1, 2, 4, 7,  9, 10, 11;
1, 2, 4, 7, 10, 12, 13, 14;
1, 2, 4, 8, 12, 15, 17, 18, 19;
1, 2, 4, 8, 13, 17, 20, 22, 23, 24;
...

Michael De Vlieger, Table of n, a(n) for n = 1..3240 (rows 1 <= n <= 80, flattened).
Sara Billey, Matjaž Konvalinka, and Joshua P. Swanson, Tableaux posets and the fake degrees of coinvariant algebras, arXiv:1809.07386 [math.CO], 2018.

Cf. A000009, A026820, A008302.

Array[Table[Count[#, ?(# <= k &)], {k, Max@ #}] &@ DeleteCases[Map[Boole[Flatten@ MapAt[Union, TakeDrop[#, LengthWhile[#, # == 1 &]], -1] == # &@ Reverse@ #] Max@ # &, Reverse@ IntegerPartitions[#]], 0] &, 13] // Flatten (* _Michael De Vlieger, Dec 12 2018 *)

A382241 Triangle read by rows: T(n,k) is the number of partitions of a 4-colored set of n objects into at most k parts with 0 <= k <= n.

Original entry on oeis.org

1, 0, 4, 0, 10, 20, 0, 20, 60, 80, 0, 35, 170, 270, 305, 0, 56, 396, 816, 1016, 1072, 0, 84, 868, 2238, 3188, 3538, 3622, 0, 120, 1716, 5616, 9196, 10996, 11556, 11676, 0, 165, 3235, 13140, 24975, 32400, 35445, 36285, 36450, 0, 220, 5720, 28900, 63680, 90700, 104060, 108820, 110020, 110240

Offset: 0

Peter Dolland, Mar 19 2025

Two unrestricted unary predicates on the n objects mean four colors: The intersection, the both differences, and the complement of the union.
The 1-color case is Euler's table A026820.
The 2-color case is A381891.
The 3-color case is A382045.

			Triangle starts:
 0 : [1]
 1 : [0,   4]
 2 : [0,  10,   20]
 3 : [0,  20,   60,    80]
 4 : [0,  35,  170,   270,    305]
 5 : [0,  56,  396,   816,   1016,   1072]
 6 : [0,  84,  868,  2238,   3188,   3538,   3622]
 7 : [0, 120, 1716,  5616,   9196,  10996,  11556,  11676]
 8 : [0, 165, 3235, 13140,  24975,  32400,  35445,  36285,  36450]
 9 : [0, 220, 5720, 28900,  63680,  90700, 104060, 108820, 110020, 110240]
10 : [0, 286, 9752, 60232, 154262, 242254, 294140, 315980, 323000, 324650, 324936]
...

Main diagonal gives A255050.

Cf. A026820, A381891, A382045, A000292.

b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1))*binomial(i*(i^2+6*i+11)/6+j, j)*x^j, j=0..n/i))))
    end:
T:= proc(n, k) option remember;
     `if`(k<0, 0, T(n, k-1)+coeff(b(n$2), x, k))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Mar 19 2025

from sympy import binomial
from sympy.utilities.iterables import partitions
from sympy.combinatorics.partitions import IntegerPartition
colors = 4 - 1   # the number of colors - 1
def a382241_row( n):
    if n == 0 : return [1]
    t = list( [0] * n)
    for p in partitions( n):
        p = IntegerPartition( p).as_dict()
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= binomial( binomial( k + colors, colors) + p[k] - 1, p[k])
        if s > 0 :
            t[s - 1] += fact
    for i in range( n - 1):
        t[i+1] += t[i]
    return [0] + t

T(n,1) = binomial(n + 3, 3) = A000292(n + 1) for n >= 1.

A383353 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where n 2-colored objects are distributed into k containers of two kinds. Containers may be left empty.

Original entry on oeis.org

1, 2, 0, 3, 4, 0, 4, 8, 6, 0, 5, 12, 22, 8, 0, 6, 16, 38, 40, 10, 0, 7, 20, 54, 92, 73, 12, 0, 8, 24, 70, 144, 196, 112, 14, 0, 9, 28, 86, 196, 354, 376, 172, 16, 0, 10, 32, 102, 248, 512, 760, 678, 240, 18, 0, 11, 36, 118, 300, 670, 1200, 1554, 1136, 335, 20, 0

Offset: 0

Peter Dolland, Apr 24 2025

			Array starts:
 0 : [1,  2,   3,    4,     5,     6,     7,      8,      9,     10,     11, ...]
 1 : [0,  4,   8,   12,    16,    20,    24,     28,     32,     36,     40, ...]
 2 : [0,  6,  22,   38,    54,    70,    86,    102,    118,    134,    150, ...]
 3 : [0,  8,  40,   92,   144,   196,   248,    300,    352,    404,    456, ...]
 4 : [0, 10,  73,  196,   354,   512,   670,    828,    986,   1144,   1302, ...]
 5 : [0, 12, 112,  376,   760,  1200,  1640,   2080,   2520,   2960,   3400, ...]
 6 : [0, 14, 172,  678,  1554,  2640,  3810,   4980,   6150,   7320,   8490, ...]
 7 : [0, 16, 240, 1136,  2936,  5436,  8272,  11228,  14184,  17140,  20096, ...]
 8 : [0, 18, 335, 1826,  5315, 10674, 17216,  24262,  31473,  38684,  45895, ...]
 9 : [0, 20, 440, 2812,  9136, 19984, 34192,  50248,  67024,  84020, 101016, ...]
10 : [0, 22, 578, 4186, 15188, 36024, 65512, 100488, 138188, 176878, 215854, ...]
...

Alois P. Heinz, Rows n = 0..200, flattened

Antidiagonal sums give A161870.
Cf. A383351, A383352.
Cf. A382345 (1-color), A381891 (1-kind), A026820 (1-color, 1-kind).
Cf. A278710.

b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, (n+1)*x^n,
      add(b(n-i*j, min(n-i*j, i-1))*binomial(i+j, j)*x^j, j=0..n/i)))
    end:
g:= proc(n, k) option remember;
      `if`(k<0, 0, g(n, k-1)+coeff(b(n$2), x, k))
    end:
A:= (n, k)-> add(add(g(j, h)*g(n-j, k-h), h=0..k), j=0..n):
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, May 05 2025

from sympy import binomial
from sympy.utilities.iterables import partitions
def calc_w( k , m):
    s = 0
    for p in partitions( m, m=k+1):
        fact = 1
        j = k + 1
        for x in p :
            fact *= binomial( j, p[x]) * (x + 1) ** p[x]
            j -= p[x]
        s += fact
    return s
def a_row( n, length=11):
    if n == 0 : return [ k + 1 for k in range( length) ]
    t = list( [0] * length)
    for p in partitions( n):
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= calc_w( k, p[k])
        if s > 0 :
            t[s - 1] += fact
    t = [0] + t
    for i in range( 1, length):
        t[i+1] += t[i] * 2 - t[i - 1]
    return t

A(0,k) = k + 1.
A(1,k) = 4*k.
A(2,k+1) = 6 + 16 * k.
A(n,1) = 2 + 2 * n.
A(n,n+k) = A(n,n) + k * A383352(n,n).
A(n,k) = Sum_{i=0..k} (k + 1 - i) * A383351(n,i) for 0 <= k <= n.
Sum_{k=0..n} (-1)^k*T(n-k,k) = A278710(n). - Alois P. Heinz, May 05 2025

A140652 Partial sums of A062968.

Original entry on oeis.org

1, 2, 4, 6, 10, 13, 19, 24, 31, 38, 48, 55, 67, 78, 90, 102, 118, 131, 149, 164, 182, 201, 223, 240, 263, 286, 310, 333, 361, 384, 414, 441, 471, 502, 534, 562, 598, 633, 669, 702, 742, 777, 819, 858, 898, 941, 987, 1026, 1073, 1118, 1166, 1213, 1265, 1312

Offset: 1

R. J. Mathar, Jul 09 2008

A062968(n) counts fractions of the format i/j with 1<=j

The partial sum gives the number of "essentially" distinct values on the unit circle for all roots up to the n-th. This relates to the problem of decomposing the generating function of the restricted partitions of n, A026820, into partial fractions.

			A062968(1)=1 counts the fraction 0/1.
A062968(2)=1 counts 1/2.
A062968(3)=2 counts {1/3,2/3}.
A062968(4)=2 counts {1/4,3/4} skipping 2/4 which could be reduced to 1/2.
A062968(5)=4 counts {1/5,2/5,3/5,4/5}. The value a(5)=1+1+2+2+4=10 counts all these distinct fractions {0/1,1/2,1/3,2/3,..,4/5}, which represent the phases of the roots of the polynomials 1-x^j, j=1..5.

Eric Weisstein's World of Mathematics, Cyclotomic Polynomial.

Cf. A062968, A007305, A002088.

Table[n + 1 - DivisorSigma[0, n], {n, 1, 54}] // Accumulate (* Jean-François Alcover, Jun 24 2013 *)

A062968(n)={ return(n+1-numdiv(n)) ; }
A(n)={ return(sum(i=1,n,A062968(i))) ; }
{ for(n=1,100,print1(A(n),", ")) ; }

a(n) = Sum_{i=1..n} A062968(i).
a(n) = Sum_{i=1..n} i - floor(n/(i+1)). - Wesley Ivan Hurt, Sep 13 2017
G.f.: x*(2 - x)/(1 - x)^3 - (1/(1 - x))*Sum_{k>=1} x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 18 2017

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cp's OEIS Frontend

A058400 Triangle of partial row sums of partition triangle A058398.

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A171985 Number of partitions of 2*n-1 into parts not greater than n.

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A233322 Triangle read by rows: T(n,k) = number of palindromic partitions of n in which no part exceeds k, 1 <= k <= n.

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A382041 Triangle read by rows: T(n, k) is the number of partitions of n with at most k parts where 0 <= k <= n, and each part is one of four kinds.

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A383352 Triangle read by rows: T(n, k) is the number of partitions of a 2-colored set of n objects into at most k parts where 0 <= k <= n, and each part is one of 2 kinds.

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A106254 Partition table in square format.

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A318806 Triangular array read by rows, where T(n,k) is the number of almost distinct partitions of n in which every part is <= k for 1 <= k <= n.

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A382241 Triangle read by rows: T(n,k) is the number of partitions of a 4-colored set of n objects into at most k parts with 0 <= k <= n.

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