A365630 -id:A365630 - cp's OEIS frontend

A060177 Triangle of generalized sum of divisors function, read by rows.

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

Offset: 1

N. J. A. Sloane, Mar 20 2001

Lengths of rows are 1 1 2 2 2 3 3 3 3 4 4 4 4 4 ... (A003056).

			Triangle turned on its side begins:
  1  2  2  3  2  4  2  4  3  4  2  6 ...
        1  2  5  6 11 13 17 22 27 29 ...
                 1  2  5 10 15 25 37 ...
                             1  2  5 ...

Alois P. Heinz, Rows n = 1..500, flattened
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.

Diagonals give A000005, A002133, A002134, A365630, A365631. Cf. A060043, A060044.

Cf. A116608 (reflected rows). - Alois P. Heinz, Jan 29 2014

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      expand(b(n, i-1) +x*add(b(n-i*j, i-1), j=1..n/i))))
    end:
T:= n->(p->seq(coeff(p, x, degree(p)-k), k=0..degree(p)-1))(b(n$2)):
seq(T(n), n=1..25);  # Alois P. Heinz, Jan 29 2014

Reverse /@ Table[Length /@ Split[ Sort[Map[Length, Split /@ IntegerPartitions[n], {1}]]], {n, 24}] (* Wouter Meeussen, Apr 21 2012, updated by Jean-François Alcover, Jan 29 2014 *)

from math import isqrt
from itertools import count, islice
from sympy.utilities.iterables import partitions
def A060177_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(isqrt((n<<3)+1)-1>>1,0,-1))
A060177_list = list(islice(A060177_gen(),30)) # Chai Wah Wu, Sep 15 2023

T(n,k) = Partitions of n using only k types of piles. Also, Sum_{k=1..A003056(n)} T(n,k)*k = A000070(n). Also, Sum_{k=1..A003056(n)} T(n,k)*(k-1) = A058884(n). - Naohiro Nomoto, Jan 24 2002

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)) = Sum_n T(n, k)*q^n.

More terms from Naohiro Nomoto, Jan 24 2002

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

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

A365664 Expansion of Sum_{0

Original entry on oeis.org

1, 3, 9, 22, 51, 97, 188, 330, 568, 918, 1452, 2233, 3344, 4884, 7004, 9856, 13653, 18699, 25080, 33462, 43918, 57304, 73668, 94482, 119262, 150285, 187231, 232560, 285660, 350746, 425627, 516477, 620731, 745503, 887796, 1056669, 1247521, 1472460, 1726054, 2021327

Offset: 10

Seiichi Manyama, Sep 15 2023

nonn

Number of partitions of n with four designated summands. For example: a(11) = 3 because there are three partitions of 11 with four designated summands: [5'+ 3'+ 2'+ 1'], [4'+ 3'+ 2'+ 1'+ 1], [4'+ 3'+ 2'+ 1 + 1']. - Omar E. Pol, Jul 26 2025

Seiichi Manyama, Table of n, a(n) for n = 10..10000
George E. Andrews and Simon C. F. Rose, MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms, Journal für die reine und angewandte Mathematik (Crelles Journal), Vol. 2013, No. 676 (2013), pp. 97-103; arXiv preprint, arXiv:1010.5769 [math.NT], 2010.

A diagonal of A060043.
Cf. A000203, A002127, A002128, A365665.
Cf. A010816, A365630, A365666.
Cf. A001158, A001160, A013955.
Column k=4 of A385001.

a[n_] := Module[{d = DivisorSigma[{1, 3, 5, 7}, n]}, (5*d[[4]] - (126*n-441)*d[[3]] + (756*n^2-4410*n+4935)*d[[2]] - (840*n^3-5880*n^2+9870*n-3229)*d[[1]])/967680]; Array[a, 40, 10] (* Amiram Eldar, Jan 07 2025 *)

a(n) = (5*sigma(n, 7)-(126*n-441)*sigma(n, 5)+(756*n^2-4410*n+4935)*sigma(n, 3)-(840*n^3-5880*n^2+9870*n-3229)*sigma(n))/967680; \\ Seiichi Manyama, Jul 24 2024

G.f.: (1/9) * ( Sum_{k>=4} (-1)^k * (2*k+1) * binomial(k+4,8) * q^(k*(k+1)/2) ) / ( Sum_{k>=0} (-1)^k * (2*k+1) * q^(k*(k+1)/2) ).
a(n) = (5*sigma_7(n) - (126*n-441)*sigma_5(n) + (756*n^2-4410*n+4935)*sigma_3(n) - (840*n^3-5880*n^2+9870*n-3229)*sigma(n))/967680. - Seiichi Manyama, Jul 24 2024
Sum_{k=1..n} a(k) ~ Pi^8 * n^8 / (8!*9!). - Vaclav Kotesovec, Aug 01 2025

A364793 Number of partitions of n with at most four part sizes.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 175, 229, 292, 375, 470, 591, 733, 905, 1103, 1343, 1615, 1938, 2309, 2726, 3211, 3758, 4379, 5069, 5865, 6716, 7694, 8769, 9967, 11254, 12732, 14264, 16025, 17877, 19959, 22149, 24605, 27147, 30012, 33006, 36294, 39742, 43573, 47524

Offset: 0

Seiichi Manyama, Sep 14 2023

nonn

Cf. A265250, A309058, A364809.

Cf. A116608, A365630.

from sympy.utilities.iterables import partitions
def A364793(n): return sum(1 for p in partitions(n) if len(p)<=4) # Chai Wah Wu, Sep 14 2023

a(n) = Sum_{k=1..4} A116608(n,k).

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A060177 Triangle of generalized sum of divisors function, read by rows.

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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.

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

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A365664 Expansion of Sum_{0

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A364793 Number of partitions of n with at most four part sizes.

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