A060177 -id:A060177 - cp's OEIS frontend

A060184 Triangle of generalized sum of divisors function, read by rows.

1, 0, 1, 2, 0, -1, 1, 2, 1, 2, 0, 0, 1, 2, 1, 1, -2, 0, 1, 3, 1, 5, 6, 0, 0, -1, -1, 2, 1, 5, 5, -2, 0, -2, -3, 2, 2, 9, 10, 0, 1, 4, 3, 0, 4, 0, 2, 9, 9, -3, 1, 3, -2, -7, 2, 0, 3, 14, 16, 0, 2, 6, -1, -9, 2, 0, 3, 15, 17, -2, 1, 8, 19, 10, -6, 4, 0, -1, 0, 15, 22, 0, 1, 9, 21, 7, -13, 2, 0, -2, -4, 11, 20, -4, 2, 15, 33, 14, -15, 3, 0, -4, -10, 10, 28, 0, 3

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  0  2 -1  2  0  2 -2  3  0  2 ...
        1  0  1  2  1  1  1  6 -1 ...
              1  0  1  0  5 -1  5 ...

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 A048272, A060185, A060186.

Cf. A060043, A060044, A060047, A060177.

max = 27(*rows*); t[n_, k_] := Module[{m, mm, q, s}, mm = Array[m, k]; s = Sum[q^Total[mm]/Times @@ (1+q^mm), Evaluate[Sequence @@ Transpose[{mm, Join[{1}, Most[mm]+1], max-Range[k-1, 0, -1]}]]]; SeriesCoefficient[s, {q, 0, n}]]; Table[Print[an = Table[t[n, k], {k, Floor[(Sqrt[8*n+1]-1)/2], 1, -1}]]; an, {n, 1, max}] // Flatten (* Jean-François Alcover, Jan 21 2014 *)

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 Vladeta Jovovic, Sep 20 2007

A092306 Number of partitions of n such that the set of parts has an even number of elements.

Original entry on oeis.org

1, 0, 0, 1, 2, 5, 6, 11, 13, 17, 23, 29, 34, 47, 64, 74, 107, 136, 185, 233, 308, 392, 518, 637, 814, 1002, 1272, 1560, 1912, 2339, 2863, 3475, 4212, 5123, 6147, 7398, 8935, 10734, 12843, 15464, 18382, 22041, 26249, 31326, 37213, 44273, 52375, 62103, 73376

Offset: 0

Vladeta Jovovic, Feb 12 2004

			The partitions of five are: {{5}, {4, 1}, {3, 2}, {3, 1, 1}, {2, 2, 1}, {2, 1, 1, 1}, {1, 1, 1, 1, 1}}, The seven partitions have 1, 2, 2, 2, 2, 2 and 1 distinct parts respectively.
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}, six of them have an even number of elements, therefore a(6)=6.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A060177, A002133, A027187, A090794.

import Data.List (group)
a092306 = length . filter even . 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

b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
       b(n, i-1, t) +add(b(n-i*j, i-1, 1-t), j=1..n/i)))
    end:
a:= n-> b(n, n, 1):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 29 2014

f[n_] := Count[ Mod[ Length /@ Union /@ IntegerPartitions[n], 2], 0]; Table[ f[n], {n, 0, 49}] (* Robert G. Wilson v, Feb 16 2004, updated by Jean-François Alcover, Jan 29 2014 *)

a(n) = b(n, 1, 0, 1) 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 main diagonal entries in the 2 X 2 matrix Product_{n >= 1} [1, x^n/(1 - x^n); x^n/(1 - x^n), 1] = [A(x), B(x); B(x), A(x)], where B(x) is the g.f. of A090794. - Peter Bala, Feb 10 2021

More terms from Robert G. Wilson v, Feb 16 2004

A067627 Triangle T(n,k) = number of conjugacy classes of partitions of n using only k types of piles, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 1, 1, 3, 2, 1, 6, 1, 3, 7, 2, 5, 9, 2, 1, 8, 11, 2, 1, 13, 14, 1, 3, 19, 15, 3, 5, 27, 19, 1, 11, 34, 22, 2, 1, 15, 49, 23, 2, 1, 27, 59, 28, 3, 3, 39, 78, 30, 1, 5, 60, 93, 34, 3, 11, 82, 118, 36, 1, 18, 115, 140, 41, 3, 1, 30, 155, 170, 42, 2, 1, 48

Offset: 1

Naohiro Nomoto, Feb 02 2002

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:
1.1.1.2.1.2.1.2.2..2..1..3..1..2..2....etc A038548
....1.1.3.3.6.7.9.11.14.15.19.22.23....etc A270060
..........1.1.3.5..8.13.19.27.34.49....etc
...................1..1..3..5.11.15....etc

Cf. A000700, A000701, A046682, A060177. Diagonals give A038548. row sums give A046682.

compareL := proc(L1,L2)
    if nops(L1) < nops(L2) then
        -1 ;
    elif nops(L1) > nops(L2) then
        1;
    else
        for i from 1 to nops(L1) do
            if op(i,L1) > op(i,L2) then
                return 1 ;
            elif op(i,L1) < op(i,L2) then
                return -1 ;
            end if;
        end do:
        0 ;
    end if;
end proc:
A067627 := proc(n,k)
    local a,p,s,pc ;
    a := 0 ;
    for p in combinat[partition](n) do
        s := convert(p,set) ;
        if nops(s) = k then
            pc := combinat[conjpart](p) ;
            if compareL(p,pc) <= 0 then
                a := a+1 ;
            end if;
        end if;
    end do:
    a ;
end proc:
for n from 1 to 30 do
for k from A003056(n) to 1 by -1 do
    printf("%4d,",A067627(n,k)) ;
end do:
printf("\n") ;
end do: # R. J. Mathar, May 08 2019

More terms from R. J. Mathar, May 08 2019

A261555 Triangle read by rows: T(n,k) is number of partitions of n having at least k distinct parts (n >= 1, k >= 1).

Original entry on oeis.org

1, 2, 3, 1, 5, 2, 7, 5, 11, 7, 1, 15, 13, 2, 22, 18, 5, 30, 27, 10, 42, 38, 16, 1, 56, 54, 27, 2, 77, 71, 42, 5, 101, 99, 62, 10, 135, 131, 87, 20, 176, 172, 128, 31, 1, 231, 226, 171, 54, 2, 297, 295, 236, 82, 5, 385, 379, 311, 127, 10, 490, 488, 417, 182, 20

Offset: 1

Michel Marcus, Aug 24 2015

From Omar E. Pol, Sep 14 2016: (Start)
Row n has length A003056(n) hence the first element of column k is in row A000217(k).
Row sums give A000070.
Alternating row sums give A090794.
Column 1 is A000041, n >= 1. (End)
[0, 0] together with column 2 gives A144300. - Omar E. Pol, Sep 17 2016

			Triangle starts:
1;
2;
3,  1;
5,  2;
7,  5;
11, 7,  1;
15, 13, 2;
22, 18, 5;
30, 27, 10;
42, 38, 16, 1;
56, 54, 27, 2;
77, 71, 42, 5;
...

Jacques Barbot, Essai sur la structuration de l'analyse combinatoire, Paris, Dulac, 1973, Annexe 12A, p. 74.

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

Cf. A000005, A000041, A000070, A000217, A003056, A060177, A090794, A116608, A144300.

Table[DeleteCases[Map[Count[Map[Length@ Union@ # &, IntegerPartitions@ n], k_ /; k >= #] &, Range@ n], 0], {n, 19}] // Flatten (* Michael De Vlieger, Sep 14 2016 *)

T(n,k) = Sum_{j>=k} A116608(n,j) assuming A116608(n,j)=0 when j>A003056(n).

T(n,1) - T(n,2) = A000005(n). - Omar E. Pol, Sep 17 2016

More terms from Alois P. Heinz, Aug 24 2015

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A060184 Triangle of generalized sum of divisors function, read by rows.

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A092306 Number of partitions of n such that the set of parts has an even number of elements.

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A067627 Triangle T(n,k) = number of conjugacy classes of partitions of n using only k types of piles, read by rows.

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A261555 Triangle read by rows: T(n,k) is number of partitions of n having at least k distinct parts (n >= 1, k >= 1).

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