A060177 -id:A060177 - cp's OEIS frontend

A104575 Alternating sum of diagonals in A060177.

1, -1, -2, -1, -1, 3, 1, 7, 4, 4, 4, 2, -9, -7, -7, -28, -17, -25, -15, -24, -11, -8, 34, 19, 53, 46, 108, 110, 106, 113, 122, 108, 75, 103, -16, -87, -107, -169, -329, -257, -574, -501, -676, -609, -749, -588, -808, -548, -521, -315, -240, 369, 485, 865, 1099, 1738, 2129, 2686, 3088, 3460, 4103, 4011, 4480, 3983

Offset: 0

Vladeta Jovovic, Apr 21 2005

A090794(n) = (A000041(n)-a(n))/2. A092306(n) = (A000041(n)+a(n))/2.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Convolution inverse of A006951.

Cf. A000041, A008965, A081362, A092306, A268498.

CoefficientList[Series[Product[(1-2x^k)/(1-x^k),{k,70}],{x,0,70}],x] (* Harvey P. Dale, Jan 21 2021 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, 1-x^k/(1-x^k))) \\ Seiichi Manyama, Oct 05 2019

G.f.: Product_{i>0} (1 - 2*x^i)/(1 - x^i).

Euler transform of -A008965(n).

a(0)=1 prepended by Seiichi Manyama, Oct 05 2019

A116608 Triangle read by rows: T(n,k) is number of partitions of n having k distinct parts (n>=1, k>=1).

Original entry on oeis.org

1, 2, 2, 1, 3, 2, 2, 5, 4, 6, 1, 2, 11, 2, 4, 13, 5, 3, 17, 10, 4, 22, 15, 1, 2, 27, 25, 2, 6, 29, 37, 5, 2, 37, 52, 10, 4, 44, 67, 20, 4, 44, 97, 30, 1, 5, 55, 117, 52, 2, 2, 59, 154, 77, 5, 6, 68, 184, 117, 10, 2, 71, 235, 162, 20, 6, 81, 277, 227, 36, 4, 82, 338, 309, 58, 1

Offset: 1

Emeric Deutsch, Feb 19 2006

Row n has floor([sqrt(1+8n)-1]/2) terms (number of terms increases by one at each triangular number).
Row sums yield the partition numbers (A000041).
Row n has length A003056(n), hence the first element of column k is in row A000217(k). - Omar E. Pol, Jan 19 2014

			T(6,2) = 6 because we have [5,1], [4,2], [4,1,1], [3,1,1,1], [2,2,1,1] and [2,1,1,1,1,1] ([6], [3,3], [3,2,1], [2,2,2] and [1,1,1,1,1,1] do not qualify).
Triangle starts:
  1;
  2;
  2,  1;
  3,  2;
  2,  5;
  4,  6, 1;
  2, 11, 2;
  4, 13, 5;
  3, 17, 10;
  4, 22, 15, 1;
  ...

Alois P. Heinz, Rows n = 1..500, flattened
Emmanuel Briand, On partitions with k corners not containing the staircase with one more corner, arXiv:2004.13180 [math.CO], 2020.
Sang June Lee and Jun Seok Oh, On zero-sum free sequences contained in random subsets of finite cyclic groups, arXiv:2003.02511 [math.CO], 2020.

Cf. A000041, A000005, A000070, A002133, A002134.
Cf. A060177 (reflected rows). - Alois P. Heinz, Jan 29 2014
Cf. A274174.

g:=product(1+t*x^j/(1-x^j),j=1..30)-1: gser:=simplify(series(g,x=0,27)): for n from 1 to 23 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 23 do seq(coeff(P[n],t^j),j=1..floor(sqrt(1+8*n)/2-1/2)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; local j; if n=0 then 1
      elif i<1 then 0 else []; for j from 0 to n/i do zip((x, y)
      ->x+y, %, [`if`(j>0, 0, [][]), b(n-i*j, i-1)], 0) od; %[] fi
    end:
T:= n-> subsop(1=NULL, [b(n, n)])[]:
seq(T(n), n=1..30); # Alois P. Heinz, Nov 07 2012
# third program
nDiffParts := proc(L)
        nops(convert(L,set)) ;
end proc:
A116608 := proc(n,k)
        local a,L;
        a :=0 ;
        for L in combinat[partition](n) do
                if nDiffParts(L) = k then
                        a := a+1 ;
                end  if;
        end do:
        a ;
end proc: # R. J. Mathar, Jun 07 2024

p=Product[1+(y x^i)/(1-x^i),{i,1,20}];f[list_]:=Select[list,#>0&];Flatten[Map[f,Drop[CoefficientList[Series[p,{x,0,20}],{x,y}],1]]] (* Geoffrey Critzer, Nov 28 2011 *)
Table[Length /@ Split[Sort[Length /@ Union /@ IntegerPartitions@n]], {n, 22}] // Flatten (* Robert Price, Jun 13 2020 *)

from math import isqrt
from itertools import count, islice
from sympy.utilities.iterables import partitions
def A116608_gen(): # generator of terms
    return (sum(1 for p in partitions(n) if len(p)==k) for n in count(1) for k in range(1,(isqrt((n<<3)+1)-1>>1)+1))
A116608_list = list(islice(A116608_gen(),30)) # Chai Wah Wu, Sep 14 2023

from functools import cache
@cache
def P(n: int, k: int, r: int) -> int:
    if n == 0: return 1 if k == 0 else 0
    if k == 0: return 0
    if r == 0: return 0
    return sum(P(n - r * j, k - 1, r - 1)
               for j in range(1, n // r + 1)) + P(n, k, r - 1)
def A116608triangle(rows: int) -> list[int]:
    return list(filter(None, [P(n, k, n) for n in range(1, rows)
                              for k in range(1, n + 1)]))
print(A116608triangle(22)) # Peter Luschny, Sep 14 2023, courtesy of Amir Livne Bar-on

G.f.: -1 + Product_{j=1..infinity} 1 + tx^j/(1-x^j).
T(n,1) = A000005(n) (number of divisors of n).
T(n,2) = A002133(n).
T(n,3) = A002134(n).
Sum_{k>=1} k * T(n,k) = A000070(n-1).
Sum_{k>=0} k! * T(n,k) = A274174(n). - Alois P. Heinz, Jun 13 2016
T(n + A000217(k), k) = A000712(n), for 0 <= n <= k [Briand]. - Álvar Ibeas, Nov 04 2020

A002133 Number of partitions of n with exactly two part sizes.

Original entry on oeis.org

0, 0, 1, 2, 5, 6, 11, 13, 17, 22, 27, 29, 37, 44, 44, 55, 59, 68, 71, 81, 82, 102, 97, 112, 109, 136, 126, 149, 141, 168, 157, 188, 176, 212, 182, 231, 207, 254, 230, 266, 241, 300, 259, 319, 283, 344, 295, 373, 311, 386, 352, 417, 353, 452, 368, 460, 418, 492, 413

Offset: 1

N. J. A. Sloane

Also number of solutions to the Diophantine equation ab + bc + cd = n, with a,b,c >= 1. - N. J. A. Sloane, Jun 17 2011

A generalized sum of divisors function.

			a(8) = 13 because we have 71, 62, 611, 53, 5111, 422, 41111, 332, 3311, 311111, 22211, 221111, 2111111.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..10000
George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. See L_3(n).
E. T. Bell, The form wx+xy+yz+zu, Bull. Amer. Math. Soc., 42 (1936), 377-380.
D. Christopher and T. Nadu, Partitions with Fixed Number of Sizes, Journal of Integer Sequences, 15 (2015), #15.11.5.
W. J. Keith, Partitions into a small number of part sizes, Int. Jour. of Num. Thy., Vol 13 no. 1, 229-241 (2017), doi:10.1142/S1793042117500130
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1921), 75-113; Coll. Papers II, pp. 303-341.
N. Benyahia Tani, S. Bouroubi, and O. Kihel, An effective approach for integer partitions using exactly two distinct sizes of parts, Bulletin du Laboratoire 03 (2015), 18-27.
N. B. Tani and S. Bouroubi, Enumeration of the Partitions of an Integer into Parts of a Specified Number of Different Sizes and Especially Two Sizes, J. Integer Seqs., Vol. 14 (2011), #11.3.6. (This sequence appears as the rightmost column of Table 1 on p. 10.)

A diagonal of A060177.

Cf. A002134.

g:=sum(sum(x^(i+j)/(1-x^i)/(1-x^j),j=1..i-1),i=1..80): gser:=series(g,x=0,65): seq(coeff(gser,x^n),n=1..60); # Emeric Deutsch, Mar 30 2006
with(numtheory); D00:=n->add(tau(j)*tau(n-j),j=1..n-1); L3:=n->(D00(n)+tau(n)-sigma(n))/2; [seq(L3(n),n=1..60)]; # N. J. A. Sloane, Jun 17 2011
A002133 := proc(n)
    A055507(n-1)+numtheory[tau](n)-numtheory[sigma](n) ;
    %/2 ;
end proc: # R. J. Mathar, Jun 15 2022
# Using function P from A365676:
A002133 := n -> P(n, 2, n): seq(A002133(n), n = 1..59); # Peter Luschny, Sep 15 2023

nn=50;ss=Sum[Sum[x^(i+j)/(1-x^i)/(1-x^j),{j,1,i-1}],{i,1,nn}];Drop[CoefficientList[Series[ss,{x,0,nn}],x],1]  (* Geoffrey Critzer, Sep 13 2012 *)
Table[DivisorSigma[0, n] - DivisorSigma[1, n] + Sum[DivisorSigma[0, k]*DivisorSigma[0, n - k], {k, 1, n - 1}], {n, 1, 100}]/2 (* Vaclav Kotesovec, Aug 30 2025 *)

from sympy import divisor_count, divisor_sigma
def A002133(n): return sum(divisor_count(j)*divisor_count(n-j) for j in range(1,(n-1>>1)+1)) + ((divisor_count(n+1>>1)**2 if n-1&1 else 0)+divisor_count(n)-divisor_sigma(n)>>1) # Chai Wah Wu, Sep 15 2023

G.f.: Sum_{i>=1} Sum_{j=1..i-1} x^(i+j)/((1-x^i)*(1-x^j)). - Emeric Deutsch, Mar 30 2006
Andrews gives a formula which is programmed up in the Maple code below. - N. J. A. Sloane, Jun 17 2011
G.f.: (G(x)^2-H(x))/2 where G(x) = Sum_{k>0} x^k/(1-x^k) and H(x) = Sum_{k>0} x^(2*k)/(1-x^k)^2. More generally, we obtain g.f. for number of partitions of n with m types of parts if we substitute x(i) with -Sum_{k>0}(x^n/(x^n-1))^i in cycle index Z(S(m); x(1),x(2),...,x(m)) of symmetric group S(m) of degree m. - Vladeta Jovovic, Sep 18 2007

A002134 Generalized divisor function. Number of partitions of n with exactly three part sizes.

Original entry on oeis.org

1, 2, 5, 10, 15, 25, 37, 52, 67, 97, 117, 154, 184, 235, 277, 338, 385, 469, 531, 630, 698, 810, 910, 1038, 1144, 1295, 1425, 1577, 1741, 1938, 2089, 2301, 2505, 2700, 2970, 3189, 3444, 3703, 4004, 4242, 4617, 4882, 5244, 5558, 5999, 6221, 6755, 7050, 7576

Offset: 6

N. J. A. Sloane

			a(8) = 5 because we have 5+2+1, 4+3+1, 4+2+1+1, 3+2+2+1, 3+2+1+1+1.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 6..10000
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

A diagonal of A060177.

Column k=3 of A116608. - Alois P. Heinz, Nov 07 2012

# Using function P from A365676:
A002134 := n -> P(n, 3, n): seq(A002134(n), n = 6..54); # Peter Luschny, Sep 15 2023

nn=40;sss=Sum[Sum[Sum[x^(i+j+k)/(1-x^i)/(1-x^j)/(1-x^k),{k,1,j-1}], {j,1,i-1}], {i,1,nn}]; Drop[CoefficientList[Series[sss,{x,0,nn}],x],6]  (* Geoffrey Critzer, Sep 13 2012 *)

G.f.: Sum_{i>=1} Sum_{j=1..i-1} Sum_{k=1..j-1} x^(i+j+k)/((1-x^i)*(1-x^j)* (1-x^k)). - Geoffrey Critzer, Sep 13 2012

Better description and more terms from Naohiro Nomoto, Jan 24 2002

More terms from Vladeta Jovovic, Nov 02 2003

A060044 Triangle of generalized sum of divisors function, read by rows.

Original entry on oeis.org

1, -1, 1, 4, -1, -5, 1, 6, 1, 3, -4, -1, -2, 8, 1, 1, -13, -2, -5, 13, 1, 10, 23, -6, -1, -11, -25, 12, 1, 12, 27, -20, -2, -21, -49, 14, 3, 31, 74, -8, 1, 5, -13, -62, 24, -1, -4, 23, 85, -29, 1, 2, -42, -132, 18, -2, -8, 42, 165, -13, 3, 14, -42, -195, 20, -4, -20, 43, 229, -30

Offset: 1

N. J. A. Sloane, Mar 19 2001

Lengths of rows are 1 1 2 2 2 3 3 3 3 ... (A003056).

			Triangle turned on its side begins:
  1  -1   4  -5   6  -4   8 -13  13 ...
          1  -1   1   3  -2   1  -5 ...
                      1  -1   1  -2 ...
For example, T(8,3) = 1.

P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

Diagonals give A002129, A002130, A060045. Cf. A060043, A060177.

Cf. A003056.

T(n, k) = sum of (-1)^(k+s_1+s_2+...+s_k) * s_1*s_2*...*s_k where s_1, s_2, ..., s_k are such that s_1*m_1 + s_2*m_2 + ... + s_k*m_k = n and the sum is over all such k-partitions of n.

G.f. for k-th diagonal (the k-th row of the sideways triangle shown in the example): Sum_{ m_1 < m_2 < ... < m_k} q^(m_1+m_2+...+m_k)/((1+q^m_1)*(1+q^m_2)*...*(1+q^m_k))^2 = Sum_n T(n, k)*q^n.

More terms from Naohiro Nomoto, Jan 24 2002

A060047 Triangle of generalized sum of divisors function, read by rows.

Original entry on oeis.org

1, 2, 4, 1, 4, 2, 6, 4, 8, 8, 8, 14, 8, 1, 18, 13, 2, 28, 12, 4, 40, 12, 8, 52, 16, 14, 70, 14, 24, 88, 16, 40, 104, 24, 1, 56, 140, 16, 2, 84, 168, 18, 4, 122, 196, 26, 8, 168, 240, 20, 14, 232, 278, 24, 24, 312, 320, 32, 40, 408, 380, 24, 64, 528, 440, 24, 100, 672, 504

Offset: 1

N. J. A. Sloane, Mar 19 2001

Lengths of rows are 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 ... (A000196).

			Triangle turned on its side begins:
  1  2  4  4  6  8  8  8 13 12 12 ...
           1  2  4  8 14 18 28 40 ...
                          1  2  4 ...
For example, T(6,1) = 8, T(6,2) = 4.

G. E. Andrews and S. C. F. Rose, MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms, arXiv:1010.5769 [math.NT], 2010.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

Diagonals give A002131, A002132, A060046, A365666, A365667.

Cf. A060043, A060044, A060177, A060184.

T(n, k) = sum of s_1*s_2*...*s_k where s_1, s_2, ..., s_k are such that s_1*(2*m_1-1) + s_2*(2*m_2-1) + ... + s_k*(2*m_k-1) = n and the sum is over all such k-partitions of n.
G.f. for k-th diagonal (the k-th row of the sideways triangle shown in the example): Sum_{ m_1 < m_2 < ... < m_k} q^(2*m_1+2*m_2+...+2*m_k-k)/((1-q^{2*m_1-1})*(1-q^{2*m_2-1})*...*(1-q^{2*m_k-1}))^2 = Sum_n T(n, k)*q^n.
G.f. for k-th diagonal: (-1)^k * (1/k) * ( Sum_{j>=k} (-1)^j * j * binomial(j+k-1,2*k-1) * q^(j^2) ) / ( 1 + 2 * Sum_{j>=1} (-q)^(j^2) ). - Seiichi Manyama, Sep 15 2023

More terms from Naohiro Nomoto, Jan 24 2002

A090794 Number of partitions of n such that the number of different parts is odd.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 4, 9, 13, 19, 27, 43, 54, 71, 102, 124, 161, 200, 257, 319, 400, 484, 618, 761, 956, 1164, 1450, 1806, 2226, 2741, 3367, 4137, 5020, 6163, 7485, 9042, 10903, 13172, 15721, 18956, 22542, 26925, 31935, 37962, 44861, 53183, 62651

Offset: 1

Vladeta Jovovic, Feb 12 2004

			n=6 has A000041(6)=11 partitions: 6, 5+1, 4+2, 4+1+1, 3+3, 3+2+1, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1 and 1+1+1+1+1+1 with partition sets: {6}, {1,5}, {2,4}, {1,4}, {3}, {1,2,3}, {1,3}, {2}, {1,2}, {1,2} and {1}, five of them have an odd number of elements, therefore a(6)=5.

Cf. A060177, A002134, A000005, A027193, A090794.
Cf. also A092314, A092322, A092269, A092309, A092321, A092313, A092310, A092311, A092268
Cf. A092306.

import Data.List (group)
a090794 = length . filter odd . map (length . group) . ps 1 where
   ps x 0 = [[]]
   ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
-- Reinhard Zumkeller, Dec 19 2013

a(n) = b(n, 1, 0, 0) with b(n, i, j, f) = if iReinhard Zumkeller, Feb 19 2004
G.f.: F(x)*G(x)/2, where F(x) = 1-Product(1-2*x^i, i=1..infinity) and G(x) = 1/Product(1-x^i, i=1..infinity).
a(n) = (A000041(n)-A104575(n))/2.
G.f. A(x) equals the off-diagonal entries in the 2 X 2 matrix Product_{n >= 1} [1, x^n/(1 - x^n); x^n/(1 - x^n), 1] = [B(x), A(x); A(x), B(x)], where B(x) is the g.f. of A092306. - Peter Bala, Feb 10 2021

More terms from Reinhard Zumkeller, Feb 17 2004

Definition simplified and shortened by Jonathan Sondow, Oct 13 2013

A365676 Triangle read by rows: T(n, k) is the number of partitions of n having exactly k distinct part sizes, for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 3, 2, 0, 0, 0, 2, 5, 0, 0, 0, 0, 4, 6, 1, 0, 0, 0, 0, 2, 11, 2, 0, 0, 0, 0, 0, 4, 13, 5, 0, 0, 0, 0, 0, 0, 3, 17, 10, 0, 0, 0, 0, 0, 0, 0, 4, 22, 15, 1, 0, 0, 0, 0, 0, 0, 0, 2, 27, 25, 2, 0, 0, 0, 0, 0, 0, 0, 0, 6, 29, 37, 5, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Peter Luschny, Sep 15 2023

			Triangle T(n, k) starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 2,  0;
  [3] 0, 2,  1,  0;
  [4] 0, 3,  2,  0, 0;
  [5] 0, 2,  5,  0, 0, 0;
  [6] 0, 4,  6,  1, 0, 0, 0;
  [7] 0, 2, 11,  2, 0, 0, 0, 0;
  [8] 0, 4, 13,  5, 0, 0, 0, 0, 0;
  [9] 0, 3, 17, 10, 0, 0, 0, 0, 0, 0;

P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., (2) 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

Variants: A116608 (nonzero terms), A060177.

Cf. A000041 (row sums), A000005 (T(n,1)), A002133 (T(n,2)), A002134 (T(n,3)), A365630 (T(n,4)), A365631 (T(n,5)).

P := proc(n, k, r) option remember; local j;  # after Amir Livne Bar-on
  if n = 0 then return ifelse(k = 0, 1, 0) fi;
  if k = 0 or r = 0 then return 0 fi;
  add(P(n - r * j, k - 1, r - 1), j = 1..iquo(n, r)) + P(n, k, r - 1) end:
A365676row := n -> local k; seq(P(n, k, n), k = 0..n):
seq(print(A365676row(n)), n = 0..9);
# Using the generating function:
p := product(1 + t*x^j/(1 - x^j), j = 1..20):
ser := series(p, x, 20):
seq(seq(coeff(coeff(ser, x, n), t, k), k = 0..n), n = 0..9);

P[n_, k_, r_] := P[n, k, r] = Which[n == 0, If[k == 0, 1, 0], k == 0 || r == 0, 0, True, Sum[P[n-r*j, k-1, r-1], {j, 1, Quotient[n, r]}]+P[n, k, r-1]]; A365676row[n_] := Table[P[n, k, n], {k, 0, n}]; Table[A365676row[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Oct 21 2023, from 1st Maple program *)

T(n,k) = my(nb=0); forpart(p=n, if (#Set(p) == k, nb++)); nb; \\ Michel Marcus, Sep 17 2023

# after Amir Livne Bar-on
from functools import cache
@cache
def P(n: int, k: int, r: int) -> int:
    if n == 0: return 1 if k == 0 else 0
    if k == 0 or r == 0: return 0
    return sum(P(n - r * j, k - 1, r - 1)
               for j in range(1, n // r + 1)) + P(n, k, r - 1)
def A365676Row(n) -> list[int]:
    return [P(n, k, n) for k in range(n + 1)]
for n in range(10): print(A365676Row(n))

T(n, k) = [t^k][x^n] Product_{j>=1} (1 + t*x^j / (1 - x^j)).

T(n, k) = 0 for n>0 and k=0. T(n,k) = 0 for k > floor([sqrt(1+8n)-1]/2). - Chai Wah Wu, Sep 15 2023

A365630 Number of partitions of n with exactly four part sizes.

Original entry on oeis.org

1, 2, 5, 10, 20, 30, 52, 77, 117, 162, 227, 309, 414, 535, 692, 873, 1100, 1369, 1661, 2030, 2438, 2925, 3450, 4108, 4759, 5570, 6440, 7457, 8491, 9798, 11020, 12593, 14125, 15995, 17820, 20074, 22182, 24833, 27379, 30422, 33351, 36996, 40346, 44445, 48336, 53048, 57494

Offset: 10

Seiichi Manyama, Sep 13 2023

nonn

			a(11) = 2 because we have 5+3+2+1, 4+3+2+1+1.

Alois P. Heinz, Table of n, a(n) for n = 10..10000

A diagonal of A060177.
Column k=4 of A116608.
Cf. A000005, A002133, A002134, A365631.

# Using function P from A365676:
A365630 := n -> P(n, 4, n): seq(A365630(n), n = 10..56); # Peter Luschny, Sep 15 2023

from sympy.utilities.iterables import partitions
def A365630(n): return sum(1 for p in partitions(n) if len(p)==4) # Chai Wah Wu, Sep 14 2023

G.f.: Sum_{0

A365631 Number of partitions of n with exactly five part sizes.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 58, 95, 147, 222, 323, 462, 636, 889, 1184, 1584, 2060, 2686, 3403, 4353, 5433, 6768, 8319, 10230, 12363, 15011, 17943, 21467, 25403, 30044, 35231, 41294, 48002, 55718, 64328, 74086, 84880, 97071, 110607, 125692, 142313, 160728, 181112, 203438, 228124

Offset: 15

Seiichi Manyama, Sep 13 2023

nonn

			a(16) = 2 because we have 6+4+3+2+1, 5+4+3+2+1+1.

Alois P. Heinz, Table of n, a(n) for n = 15..5000

A diagonal of A060177.
Column k=5 of A116608.
Cf. A000005, A002133, A002134, A365630.
Cf. A364809.

# Using function P from A365676:
A365631 := n -> P(n, 5, n): seq(A365631(n), n = 15..59); # Peter Luschny, Sep 15 2023

from sympy.utilities.iterables import partitions
def A365631(n): return sum(1 for p in partitions(n) if len(p)==5) # Chai Wah Wu, Sep 14 2023

G.f.: Sum_{0

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cp's OEIS Frontend

A104575 Alternating sum of diagonals in A060177.

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A116608 Triangle read by rows: T(n,k) is number of partitions of n having k distinct parts (n>=1, k>=1).

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A002133 Number of partitions of n with exactly two part sizes.

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A002134 Generalized divisor function. Number of partitions of n with exactly three part sizes.

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A060044 Triangle of generalized sum of divisors function, read by rows.

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A060047 Triangle of generalized sum of divisors function, read by rows.

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A090794 Number of partitions of n such that the number of different parts is odd.

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