A026820 -id:A026820 - cp's OEIS frontend

A008636 Number of partitions of n into at most 7 parts.

1, 1, 2, 3, 5, 7, 11, 15, 21, 28, 38, 49, 65, 82, 105, 131, 164, 201, 248, 300, 364, 436, 522, 618, 733, 860, 1009, 1175, 1367, 1579, 1824, 2093, 2400, 2738, 3120, 3539, 4011, 4526, 5102, 5731, 6430, 7190, 8033, 8946, 9953, 11044, 12241, 13534, 14950, 16475, 18138

Offset: 0

N. J. A. Sloane, Mar 15 1996

Also, the number of partitions of n into parts <= 7: a(n) = A026820(n, 7). - Reinhard Zumkeller, Jan 21 2010
Counts unordered closed walks of weight n on a single vertex graph with 7 loops of weights 1, 2, 3, 4, 5, 6 and 7. - David Neil McGrath, Apr 11 2015
Number of different distributions of n+28 identical balls in 7 boxes as x,y,z,p,q,m,n where 0 < x < y < z < p < q < m < n. - Ece Uslu and Esin Becenen, Jan 11 2016

			There are 28 partitions of 9 into parts less than or equal to 7. These are (72)(711)(63)(621)(6111)(54)(531)(522)(5211)(51111)(441)(432)(4311)(4221)(42111)(411111)(333)(3321)(33111)(3222)(32211)(321111)(3111111)(22221)(222111)(2211111)(21111111)(111111111). - _David Neil McGrath_, Apr 11 2015
a(3) = 3, i.e., {1,2,3,4,5,7,9}, {1,2,3,4,6,7,8}, {1,2,3,4,5,6,10}. Number of different distributions of 31 identical balls in 7 boxes as x,y,z,p,q,m,n where 0 < x < y < z < p < q < m < n. - _Ece Uslu_, Esin Becenen, Jan 11 2016

A. Cayley, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 415.
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.

T. D. Noe, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 356
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,0,-1,-1,0,1,1,2,0,0,0,-2,-1,-1,0,1,1,0,1,0,0,-1,-1,1).

Cf. A008284, A026820.

with(combstruct):ZL8:=[S,{S=Set(Cycle(Z,card<8))}, unlabeled]: seq(count(ZL8,size=n),n=0..48); # Zerinvary Lajos, Sep 24 2007
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=7)},unlabelled]: seq(combstruct[count](B, size=n), n=0..48); # Zerinvary Lajos, Mar 21 2009

CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 7} ], {x, 0, 60} ], x ]

{a(n)=(2*n^6+168*n^5+5530*n^4+90160*n^3+754299*n^2+(2988020+44800*(1-n%3))*n+6654375+1575*(3*n^2+84*n+511)*(-1)^n)\7257600}; \\ Tani Akinari, May 27 2014

G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)).
a(n) = A008284(n+7, 7), n >= 0.
a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) - a(n-8) + a(n-10) + a(n-11) + 2*a(n-12) - 2*a(n-16) - a(n-17) - a(n-18) + a(n-20) + a(n-21) + a(n-23) - a(n-26) - a(n-27) + a(n-28). - David Neil McGrath, Apr 11 2015
a(n+7) = a(n) + A001402(n). - Ece Uslu, Esin Becenen, Jan 11 2016
a(n) = A026813(n+7). - R. J. Mathar, Feb 13 2019
From Vladimír Modrák, Jul 30 2022: (Start)
a(n) = Sum_{p=0..floor(n/7)} Sum_{m=0..floor(n/6)} Sum_{k=0..floor(n/5)} Sum_{j=0..floor(n/4)} Sum_{i=0..floor(n/3)} ceiling((max(0, n + 1 - 3*i - 4*j - 5*k - 6*m - 7*p))/2).
a(n) = Sum_{m=0..floor(n/7)} Sum_{k=0..floor(n/6)} Sum_{j=0..floor(n/5)} Sum_{i=0..floor(n/4)} floor(((max(0, n + 3 - 4*i - 5*j - 6*k - 7*m))^2+4)/12). (End)

More terms from Robert G. Wilson v, Dec 11 2000

A008637 Number of partitions of n into at most 8 parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 40, 52, 70, 89, 116, 146, 186, 230, 288, 352, 434, 525, 638, 764, 919, 1090, 1297, 1527, 1801, 2104, 2462, 2857, 3319, 3828, 4417, 5066, 5812, 6630, 7564, 8588, 9749, 11018, 12450, 14012, 15765, 17674, 19805, 22122

Offset: 0

N. J. A. Sloane

For n>7: also number of partitions of n into parts <= 8: a(n)=A026820(n,8). - Reinhard Zumkeller, Jan 21 2010
Molien series for finite Coxeter group of type A_8.
Number of different distributions of n+36 identical balls in 8 boxes as x,y,z,p,q,m,n,h where 0 < x < y < z < p < q < m < n < h. - Ece Uslu and Esin Becenen, Jan 11 2016

			There are a(9)=29 partitions of 9 into parts less than or equal to 8. These are (81)(72)(711)(63)(621)(6111)(54)(531)(522)(5211)(51111)(441)(432)(4311)(4221)(42111)(411111)(333)(3321)(33111)(3222)(32211)(321111)(3111111)(22221)(222111)(2211111)(21111111)(111111111). - _David Neil McGrath_, Apr 14 2015
a(3) = 3, i.e., {1,2,3,4,5,7,8,9}, {1,2,3,4,5,6,8,10}, {1,2,3,4,5,6,7,11}: number of different distributions of 39 identical balls in 8 boxes as x,y,z,p,q,m,n,h where 0 < x < y < z < p < q < m < n < h. - _Ece Uslu_, Esin Becenen, Jan 11 2016

A. Cayley, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 415.
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.

T. D. Noe, Table of n, a(n) for n = 0..1000
Tani Akinari, Formula for a(n)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 357
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,0,-1,0,-1,0,1,2,1,0,1,-1,-1,-2,-1,-1,1,0,1,2,1,0,-1,0,-1,0,-1,0,0,1,1,-1).

Cf. A008284.

Strictly different from A008631, although they have similar descriptions.

1/(1-x)/(1-x^2)/(1-x^3)/(1-x^4)/(1-x^5)/(1-x^6)/(1-x^7)/(1-x^8)
with(combstruct):ZL9:=[S,{S=Set(Cycle(Z,card<9))}, unlabeled]:seq(count(ZL9,size=n),n=0..47); # Zerinvary Lajos, Sep 24 2007
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=8)},unlabelled]: seq(combstruct[count](B, size=n), n=0..47); # Zerinvary Lajos, Mar 21 2009

CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 8} ], {x, 0, 60} ], x ]

a(n):=floor((-1)^n*((n+1)*(-1)^floor((n+2)/3)+(2*n+3)*(-1)^floor((n+1)/3)+(n+2)*(-1)^floor(n/3))/972+(n+2)*((-1)^n+1)*(-1)^(n/2)/512+(n+18)*(6*n^6+648*n^5+27018*n^4+545616*n^3+5481213*n^2+25163028*n+39226571)/1219276800+(n+1)*(n^2+53*n+826)*(-1)^n/36864+1/2); /* Tani Akinari, Oct 25 2012 */

G.f.: 1/((1-t)*(1-t^2)*(1-t^3)*(1-t^4)*(1-t^5)*(1-t^6)*(1-t^7)*(1-t^8)). - N. J. A. Sloane, Jan 09 2016
a(n) = A008284(n+8, 8), n >= 0.
a(n) = floor((-1)^n*((n+1)*(-1)^(floor((n+2)/3)) + (2*n+3)*(-1)^(floor((n+1)/3)) + (n+2)*(-1)^(floor(n/3)))/972 + (n+2)*((-1)^n+1)*(-1)^(n/2)/512 + (n+18)*(6*n^6 + 648*n^5 + 27018*n^4 + 545616*n^3 + 5481213*n^2 + 25163028*n + 39226571)/1219276800 + (n+1)*(n^2+53*n+826)*(-1)^n/36864+1/2). (See link.) - Tani Akinari, Oct 26 2012
a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) - a(n-9) + a(n-11) + 2*a(n-12) + a(n-13) + a(n-15) - a(n-16) - a(n-17) - 2*a(n-18) - a(n-19) - a(n-20) + a(n-21) + a(n-23) + 2*a(n-24) + a(n-25) - a(n-27) - a(n-29) - a(n-31) + a(n-34) + a(n-35) - a(n-36). - David Neil McGrath, Apr 14 2015
a(n+8) = a(n) + A008636(n). - Ece Uslu, Esin Becenen, Jan 11 2016
From Vladimír Modrák, Jul 30 2022: (Start)
a(n) = Sum_{i_1=0..floor(n/3)} Sum_{i_2=0..floor(n/4)} Sum_{i_3=0..floor(n/5)} Sum_{i_4=0..floor(n/6)} Sum_{i_5=0..floor(n/7)} Sum_{i_6=0..floor(n/8)} ceiling((max(0, n + 1 - 3*i_1 - 4*i_2 - 5*i_3 - 6*i_4 - 7*i_5 - 8*i_6))/2).
a(n) = Sum_{i_1=0..floor(n/4)} Sum_{i_2=0..floor(n/5)} Sum_{i_3=0..floor(n/6)} Sum_{i_4=0..floor(n/7)} Sum_{i_5=0..floor(n/8)} floor(((max(0, n + 3 - 4*i_1 - 5*i_2 - 6*i_3 - 7*i_4 - 8*i_5))^2+4)/12). (End)

More terms from Robert G. Wilson v, Dec 11 2000

A008638 Number of partitions of n into at most 9 parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 54, 73, 94, 123, 157, 201, 252, 318, 393, 488, 598, 732, 887, 1076, 1291, 1549, 1845, 2194, 2592, 3060, 3589, 4206, 4904, 5708, 6615, 7657, 8824, 10156, 11648, 13338, 15224, 17354, 19720, 22380, 25331, 28629, 32278

Offset: 0

N. J. A. Sloane

For n > 8: also number of partitions of n into parts <= 9: a(n) = A026820(n, 9). - Reinhard Zumkeller, Jan 21 2010

A. Cayley, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 415.
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.

T. D. Noe, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 358
Index entries for related partition-counting sequences
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,0,-1,0,0,-1,0,2,1,1,1,0, -1,-1,-1,-2,-1,-1,1,1,2,1,1,1,0, -1,-1,-1,-2,0,1,0,0,1,0,1,0,0,-1,-1,1)

Essentially same as A026815.
a(n) = A008284(n+9, 9), n >= 0.
Cf. A288344 (partial sums), A266777 (first differences).

CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 9} ], {x, 0, 60} ], x ]

G.f.: 1/Product_{k=1..9} (1 - q^k).
a(n) = floor((30*n^8 + 5400*n^7 + 405300*n^6 + 16443000*n^5 + 390533640*n^4 + 5486840100*n^3 + 43691213950*n^2 + 175052776500*n + 256697834389)/438939648000 + (n + 1)*(2*n^2 + 133*n + 2597)*(-1)^n/147456 + (-1)^n*((n + 1)*(n + 47)*(-1)^floor(n/3 + 2/3) + (2*n^2 + 90*n + 127)*(-1)^floor(n/3 + 1/3) + (n + 2)*(n + 40)*(-1)^floor(n/3))/17496 + 1/256*((-1)^((2*n + (-1)^n - 1)/4)*floor((n + 2)/2)) + 1/2). - Tani Akinari, Oct 20 2012
a(n) = a(n-9) + A008637(n). - Vladimír Modrák, Sep 28 2020
From Vladimír Modrák, Aug 09 2022: (Start)
a(n) = Sum_{i_1=0..floor(n/3)} Sum_{i_2=0..floor(n/4)} Sum_{i_3=0..floor(n/5)} Sum_{i_4=0..floor(n/6)} Sum_{i_5=0..floor(n/7)} Sum_{i_6=0..floor(n/8)} Sum_{i_7=0..floor(n/9)} ceiling((max(0, n + 1 - 3*i_1 - 4*i_2 - 5*i_3 - 6*i_4 - 7*i_5 - 8*i_6 - 9*i_7))/2).
a(n) = Sum_{i_1=0..floor(n/4)} Sum_{i_2=0..floor(n/5)} Sum_{i_3=0..floor(n/6)} Sum_{i_4=0..floor(n/7)} Sum_{i_5=0..floor(n/8)} Sum_{i_6=0..floor(n/9)} floor(((max(0, n + 3 - 4*i_1 - 5*i_2 - 6*i_3 - 7*i_4 - 8*i_5 - 9*i_6))^2+4)/12). (End)

A229915 Number of espalier polycubes of a given volume in dimension 3.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 26, 34, 57, 76, 116, 150, 227, 284, 408, 520, 718, 895, 1226, 1508, 2018, 2487, 3248, 3968, 5160, 6235, 7970, 9653, 12179, 14630, 18367, 21924, 27241, 32506, 39985, 47492, 58203, 68752, 83613, 98730, 119269, 140224, 168799, 197758, 236753, 277052, 329867, 384852, 457006, 531500, 628338

Offset: 0

Matthieu Deneufchâtel, Oct 03 2013

nonn

A pyramid polycube is obtained by gluing together horizontal plateaux (parallelepipeds of height 1) in such a way that (0,0,0) belongs to the first plateau and each cell with coordinates (0,b,c) belongs to the first plateau such that b,c >= 0. If the cell with coordinates (a,b,c) belongs to the (a+1)-st plateau (a>0), then the cell with coordinates (a-1, b, c) belongs to the a-th plateau.

An espalier polycube is a special pyramid such that each plateau contains the cell with coordinates (a,0,0).

Seiichi Manyama, Table of n, a(n) for n = 0..100
C. Carré, N. Debroux, M. Deneufchatel, J.-P. Dubernard et al., Dirichlet convolution and enumeration of pyramid polycubes, 2013.
C. Carre, N. Debroux, M. Deneufchatel, J.-Ph. Dubernard, C. Hillariet, J.-G. Luque, and O. Mallet, Enumeration of Polycubes and Dirichlet Convolutions, J. Int. Seq. 18 (2015) 15.11.4.

Cf. A026820, A100882, A227925, A230118, A229917, A229925, A323582.

The generating function for the numbers of espaliers of height h and volumes v_1 , ... v_h is x_1^{n_1} * ... x_h^{n_h} / ((1-x_1^{n_1}) *(1-x_1^{n_1}*x_2^{n_2}) *... *(1-x_1^{n_1}*x_2^{n_2}*...x_h^{n_h})).

This sequence is obtained with x_1 = ... = x_h = p by summing over n_1>= ... >= n_h>=1 and then over h.

a(0)=1 prepended by Seiichi Manyama, Aug 20 2020

A322440 Number of pairs of integer partitions of n where every part of the first is less than every part of the second.

Original entry on oeis.org

1, 0, 1, 2, 5, 7, 16, 20, 40, 55, 97, 124, 235, 287, 482, 654, 1033, 1318, 2137, 2676, 4157, 5439, 7891, 10144, 15280, 19171, 27336, 35652, 49756, 63150, 89342, 111956, 154400, 197413, 264572, 336082, 456724, 568932, 756065, 959566, 1261803, 1576355, 2078267

Offset: 0

Gus Wiseman, Dec 08 2018

nonn

			The a(5) = 16 pairs of integer partitions:
      (51)|(6)
      (42)|(6)
     (411)|(6)
      (33)|(6)
     (321)|(6)
    (3111)|(6)
     (222)|(6)
     (222)|(33)
    (2211)|(6)
    (2211)|(33)
   (21111)|(6)
   (21111)|(33)
  (111111)|(6)
  (111111)|(42)
  (111111)|(33)
  (111111)|(222)

Cf. A265947, A317144, A318915, A322435, A322436, A322439.

g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      g(n, i-1) +g(n-i, min(i, n-i)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(g(n-i, min(n-i, i))*b(n, i+1), i=1..n))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 09 2018

Table[Length[Select[Tuples[IntegerPartitions[n],2],Max@@First[#]n, 0, b[n, i+1] + b[n-i, i]]];
a[n_] := a[n] = If[n==0, 1, Sum[g[n-i, Min[n-i, i]]*b[n, i+1], {i, 1, n}]];
a /@ Range[0, 50] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

a(n) = Sum_{k=1..n-1} A026820(n, k) * A026794(n, k + 1).

A381891 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 with 0 <= k <= n.

Original entry on oeis.org

1, 0, 2, 0, 3, 6, 0, 4, 10, 14, 0, 5, 19, 28, 33, 0, 6, 28, 52, 64, 70, 0, 7, 44, 93, 127, 142, 149, 0, 8, 60, 152, 228, 272, 290, 298, 0, 9, 85, 242, 404, 507, 561, 582, 591, 0, 10, 110, 370, 672, 904, 1034, 1098, 1122, 1132, 0, 11, 146, 546, 1100, 1568, 1870, 2027, 2101, 2128, 2139

Offset: 0

Peter Dolland, Mar 09 2025

The 1-color case is Euler's table A026820.

			Triangle begins:
  1;
  0, 2;
  0, 3,  6;
  0, 4, 10,  14;
  0, 5, 19,  28,  33;
  0, 6, 28,  52,  64,  70;
  0, 7, 44,  93, 127, 142, 149;
  0, 8, 60, 152, 228, 272, 290, 298;
  ...

Alois P. Heinz, Rows n = 0..150, flattened

Cf. A005380, A026820.

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:
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 09 2025

from sympy import binomial
from sympy.utilities.iterables import partitions
from sympy.combinatorics.partitions import IntegerPartition
def a381891_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( k + p[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(1,k) = k + 1.

T(n,n) = A005380(n).

A381895 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 two kinds.

Original entry on oeis.org

1, 0, 2, 0, 2, 5, 0, 2, 6, 10, 0, 2, 9, 15, 20, 0, 2, 10, 22, 30, 36, 0, 2, 13, 31, 48, 58, 65, 0, 2, 14, 40, 68, 90, 102, 110, 0, 2, 17, 51, 97, 135, 162, 176, 185, 0, 2, 18, 64, 128, 194, 242, 274, 290, 300, 0, 2, 21, 77, 171, 271, 357, 415, 452, 470, 481

Offset: 0

Peter Dolland, Mar 09 2025

The 1-kind case is Euler's table A026820.

			Triangle starts:
   0 : [1]
   1 : [0, 2]
   2 : [0, 2,  5]
   3 : [0, 2,  6, 10]
   4 : [0, 2,  9, 15,  20]
   5 : [0, 2, 10, 22,  30,  36]
   6 : [0, 2, 13, 31,  48,  58,  65]
   7 : [0, 2, 14, 40,  68,  90, 102, 110]
   8 : [0, 2, 17, 51,  97, 135, 162, 176, 185]
   9 : [0, 2, 18, 64, 128, 194, 242, 274, 290, 300]
  10 : [0, 2, 21, 77, 171, 271, 357, 415, 452, 470, 481]
  ...

Alois P. Heinz, Rows n = 0..150, flattened

Main diagonal gives A000712.

Cf. A026820.

A381895(row_max) = {my(N=row_max+1,x='x+O('x^N), y='y+O('y^N), h=prod(i=1,N, 1/(1-y*x^i)^2)/(1-y)); for(n=0,N-1, if(n<1, print([1]),print(concat([0],Vec(polcoeff(h, n))[1..n]))))}
A381895(12) \\ John Tyler Rascoe, Mar 19 2025

from sympy.utilities.iterables import partitions
from sympy.combinatorics.partitions import IntegerPartition
def a381895_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 *= 1 + p[k]
        if s > 0 :
            t[s - 1] += fact
    for i in range( n - 1):
        t[i+1] += t[i]
    return [0] + t

G.f.: A(x,y,2) where A(x,y,p) = 1/(1-y) * Product_{i>0} 1/(1-y*x^i)^p is the g.f for the number of partitions of n with at most k parts and p kinds of each part. - John Tyler Rascoe, Mar 19 2025

A384915 The number of unordered factorizations of n into powers of primes of the form p^e where p is prime and 0 <= e <= p (A074583).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 3, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 12 2025

			a(4) = 2 since 4 has 2 factorizations: 2^1 * 2^1 and 2^2, with exponents 1 and 2 that are <= 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A026820, A048103, A074583.

Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384913, A384914, A384916.

f[p_, e_] := Length[IntegerPartitions[e, p]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

T(n, k)=my(s); forpart(v=n, s++, , k); s \\ Charles R Greathouse IV at A026820
a(n) = {my(f = factor(n)); prod(i = 1, #f~, T(f[i,2], f[i,1]));}

Multiplicative with a(p^e) = A026820(e, p).

a(n) >= A384916(n), with equality if and only if n is in A048103.

A384916 The number of unordered factorizations of n into powers of primes of the form p^e where p is prime and 0 <= e < p.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 12 2025

First differs from A298735 at n = 125.

			a(9) = 2 since 9 has 2 factorizations: 3^1 * 3^1 and 3^2, with exponents 1 and 2 that are < 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A026820, A048103, A298735.

Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384913, A384914, A384915.

f[p_, e_] := Length[IntegerPartitions[e, p-1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

T(n, k)=my(s); forpart(v=n, s++, , k); s \\ Charles R Greathouse IV at A026820
a(n) = {my(f = factor(n)); prod(i = 1, #f~, T(f[i,2], f[i,1]-1));}

Multiplicative with a(p^e) = A026820(e, p-1).

a(n) <= A384915(n), with equality if and only if n is in A048103.

A104382 Triangle, read by rows, where T(n,k) equals number of distinct partitions of triangular number n*(n+1)/2 into k different summands for n>=k>=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 7, 12, 6, 1, 1, 10, 27, 27, 10, 1, 1, 13, 52, 84, 57, 14, 1, 1, 17, 91, 206, 221, 110, 21, 1, 1, 22, 147, 441, 674, 532, 201, 29, 1, 1, 27, 225, 864, 1747, 1945, 1175, 352, 41, 1, 1, 32, 331, 1575, 4033, 5942, 5102, 2462, 598, 55, 1, 1, 38, 469

Offset: 1

Paul D. Hanna, Mar 04 2005

Secondary diagonal equals partitions of n - 1 (A000065).
Third diagonal is A104384.
Third column is A104385.
Row sums are A104383 where limit_{n --> inf} A104383(n+1)/A104383(n) = exp(sqrt(Pi^2/6)) = 3.605822247984...

			Rows begin:
1;
1, 1;
1, 2, 1;
1, 4, 4, 1;
1, 7, 12, 6, 1;
1, 10, 27, 27, 10, 1;
1, 13, 52, 84, 57, 14, 1;
1, 17, 91, 206, 221, 110, 21, 1;
1, 22, 147, 441, 674, 532, 201, 29, 1;
1, 27, 225, 864, 1747, 1945, 1175, 352, 41, 1;
1, 32, 331, 1575, 4033, 5942, 5102, 2462, 598, 55, 1; ...

Abramowitz, M. and Stegun, I. A. (Editors). "Partitions into Distinct Parts." S24.2.2 in Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9th printing. New York: Dover, pp. 825-826, 1972.

Alois P. Heinz, Rows n = 1..55, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Eric Weisstein's World of Mathematics, Partition Function Q.

Cf. A008289, A000009, A000065, A104383, A104384, A104385.

PARI
```
T(n,k)=if(n
				
```

T(n, 1) = T(n, n) = 1.
T(n, n-1) = A000065(n).
T(n, 2) = [(n*(n+1)/2-1)/2].
From Álvar Ibeas, Jul 23 2020: (Start)
T(n, k) = A008284((n-k+1)*(n+k)/2, k).
T(n, k) = A026820((n-k)*(n+k+1)/2, k), with A026820(0, k) = 1. (End)

cp's OEIS Frontend

A008636 Number of partitions of n into at most 7 parts.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A008637 Number of partitions of n into at most 8 parts.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

Formula

Extensions

A008638 Number of partitions of n into at most 9 parts.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A229915 Number of espalier polycubes of a given volume in dimension 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A322440 Number of pairs of integer partitions of n where every part of the first is less than every part of the second.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A381891 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 with 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Python

Formula

A381895 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 two kinds.

Views

Author

Keywords

Comments

Examples

Links