A332304 -id:A332304 - cp's OEIS frontend

A339442 Number of compositions (ordered partitions) of n into an odd number of distinct triangular numbers.

0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 7, 0, 0, 0, 6, 1, 0, 6, 0, 12, 0, 1, 6, 0, 6, 6, 6, 0, 13, 0, 6, 6, 12, 0, 6, 126, 1, 18, 0, 12, 6, 126, 6, 6, 12, 7, 132, 6, 120, 18, 126, 0, 24, 246, 12, 127, 126, 126, 12, 132, 126, 138, 126, 132, 12, 246, 133, 138, 366, 6, 258, 252

Offset: 0

Ilya Gutkovskiy, Dec 05 2020

nonn

			a(19) = 12 because we have [15, 3, 1] (6 permutations) and [10, 6, 3] (6 permutations).

Cf. A000217, A331843, A332304, A339376, A339417, A339441.

b:= proc(n, i, p) option remember; `if`(n=0, irem(p, 2)*p!, (t->
     `if`(t>n, 0, b(n, i+1, p)+b(n-t, i+1, p+1)))(i*(i+1)/2))
    end:
a:= n-> b(n, 1, 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 05 2020

b[n_, i_, p_] := b[n, i, p] = If[n == 0, Mod[p, 2]*p!, With[{t = i(i+1)/2}, If[t > n, 0, b[n, i + 1, p] + b[n - t, i + 1, p + 1]]]];
a[n_] := b[n, 1, 0];
a /@ Range[0, 100] (* Jean-François Alcover, Mar 14 2021, after Alois P. Heinz *)

A346634 Number of strict odd-length integer partitions of 2n + 1.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 9, 14, 19, 27, 38, 52, 71, 96, 128, 170, 224, 293, 380, 491, 630, 805, 1024, 1295, 1632, 2048, 2560, 3189, 3958, 4896, 6038, 7424, 9100, 11125, 13565, 16496, 20013, 24223, 29250, 35244, 42378, 50849, 60896, 72789, 86841, 103424, 122960, 145937

Offset: 0

Gus Wiseman, Aug 01 2021

nonn

			The a(0) = 1 through a(7) = 14 partitions:
  (1)  (3)  (5)  (7)      (9)      (11)     (13)      (15)
                 (4,2,1)  (4,3,2)  (5,4,2)  (6,4,3)   (6,5,4)
                          (5,3,1)  (6,3,2)  (6,5,2)   (7,5,3)
                          (6,2,1)  (6,4,1)  (7,4,2)   (7,6,2)
                                   (7,3,1)  (7,5,1)   (8,4,3)
                                   (8,2,1)  (8,3,2)   (8,5,2)
                                            (8,4,1)   (8,6,1)
                                            (9,3,1)   (9,4,2)
                                            (10,2,1)  (9,5,1)
                                                      (10,3,2)
                                                      (10,4,1)
                                                      (11,3,1)
                                                      (12,2,1)
                                                      (5,4,3,2,1)

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Odd bisection of A067659, which is ranked by A030059.
The even version is the even bisection of A067661.
The case of all odd parts is counted by A069911 (non-strict: A078408).
The non-strict version is A160786, ranked by A340931.
The non-strict even version is A236913, ranked by A340784.
The even-length version is A343942 (non-strict: A236914).
The even-sum version is A344650 (non-strict: A236559 or A344611).
A000009 counts partitions with all odd parts, ranked by A066208.
A000009 counts strict partitions, ranked by A005117.
A027193 counts odd-length partitions, ranked by A026424.
A027193 counts odd-maximum partitions, ranked by A244991.
A058695 counts partitions of odd numbers, ranked by A300063.
A340385 counts partitions with odd length and maximum, ranked by A340386.
Other cases of odd length:
- A024429 set partitions
- A089677 ordered set partitions
- A166444 compositions
- A174726 ordered factorizations
- A332304 strict compositions
- A339890 factorizations
Cf. A000700, A008289, A047993, A072233, A106529, A168659, A218171, A340102, A340604, A340607.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
    end:
a:= n-> b(2*n+1$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 05 2021

Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&OddQ[Length[#]]&]],{n,0,15}]

More terms from Alois P. Heinz, Aug 05 2021

A339435 G.f.: Sum_{k>=0} (-1)^k * k! * x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j).

Original entry on oeis.org

1, -1, -1, 1, 1, 3, -3, -1, -7, -11, 7, 3, 15, 35, 71, -35, 25, -57, -99, -277, -415, 25, -185, 39, 327, 1079, 1895, 3745, -71, 2907, 813, 479, -4927, -7259, -20393, -29877, -8409, -28621, -23041, -16811, 4441, 27783, 102741, 169595, 324065, 105265, 361471, 280983, 385215

Offset: 0

Ilya Gutkovskiy, Dec 04 2020

The difference between the number of compositions (ordered partitions) of n into an even number of distinct parts and the number of compositions (ordered partitions) of n into an odd number of distinct parts.

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A010815, A032020, A081362, A332304, A332305.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 0):
seq(a(n), n=0..55);  # Alois P. Heinz, Dec 04 2020

nmax = 48; CoefficientList[Series[Sum[(-1)^k k! x^(k (k + 1)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

a(n) = A332305(n) - A332304(n).

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A339442 Number of compositions (ordered partitions) of n into an odd number of distinct triangular numbers.

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A346634 Number of strict odd-length integer partitions of 2n + 1.

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A339435 G.f.: Sum_{k>=0} (-1)^k * k! * x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j).

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