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

Author

N. J. A. Sloane, Mar 20 2001

Keywords

• nonn
• tabf
• easy
• nice
• look

Comments

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

Examples

			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 ...
		

Links

• 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.

Crossrefs

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

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

Programs

Maple

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

Mathematica

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 *)

Python

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

Formula

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.

Extensions

More terms from Naohiro Nomoto, Jan 24 2002