A116674 - OEIS

A116674

Triangle read by rows: T(n,k) is the number of partitions of n into odd parts and having exactly k distinct parts (n>=1, k>=1).

1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 3, 1, 5, 3, 4, 1, 2, 7, 1, 2, 8, 2, 2, 10, 3, 2, 11, 5, 2, 13, 7, 4, 12, 11, 1, 19, 11, 1, 2, 18, 17, 1, 3, 20, 21, 2, 2, 22, 27, 3, 2, 25, 32, 5, 4, 24, 41, 7, 2, 30, 46, 11, 2, 31, 56, 15, 2, 36, 62, 22, 3, 33, 80, 25, 1, 2, 39, 87, 36, 1, 4, 38, 103, 45, 2, 2, 45

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Row n has floor(sqrt(n)) terms. Row sums yield A000009. T(n,1)=A001227(n) (n>=1). Sum(k*T(n,k),k>=1)=A038348(n-1) (n>=1).

Conjecture: Also the number of strict integer partitions of n with k maximal runs of consecutive parts decreasing by 1. - Gus Wiseman, Jun 24 2025

LINKS

Alois P. Heinz, Rows n = 1..1000, flattened

FORMULA

G.f.: product(1+tx^(2j-1)/(1-x^(2j-1)), j=1..infinity).

EXAMPLE

From Gus Wiseman, Jun 24 2025: (Start)

Triangle begins:

1: 1

2: 1

3: 2

4: 1 1

5: 2 1

6: 2 2

7: 2 3

8: 1 5

9: 3 4 1

10: 2 7 1

11: 2 8 2

12: 2 10 3

13: 2 11 5

14: 2 13 7

15: 4 12 11

16: 1 19 11 1

17: 2 18 17 1

18: 3 20 21 2

19: 2 22 27 3

20: 2 25 32 5

Row n = 9 counts the following partitions into odd parts by number of distinct parts:

(9) (7,1,1) (5,3,1)

(3,3,3) (3,3,1,1,1)

(1,1,1,1,1,1,1,1,1) (5,1,1,1,1)

(3,1,1,1,1,1,1)

Row n = 9 counts the following strict partitions by number of maximal runs:

(9) (6,3) (5,3,1)

(5,4) (7,2)

(4,3,2) (8,1)

(6,2,1)

(End)

MAPLE

g:=product(1+t*x^(2*j-1)/(1-x^(2*j-1)), j=1..35): gser:=simplify(series(g, x=0, 34)): for n from 1 to 29 do P[n]:=coeff(gser, x^n) od: for n from 1 to 29 do seq(coeff(P[n], t, j), j=1..floor(sqrt(n))) od; # yields sequence in triangular form

# Alternative:

with(numtheory):

b:= proc(n, i) option remember; expand(`if`(n=0, 1,

`if`(i<1, 0, add(b(n-i*j, i-2)*`if`(j=0, 1, x), j=0..n/i))))

end:

T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(

b(n, iquo(n+1, 2)*2-1)):

seq(T(n), n=1..30); # Alois P. Heinz, Mar 08 2015

MATHEMATICA

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-2]*If[j == 0, 1, x], {j, 0, n/i}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, Quotient[n+1, 2]*2-1]]; Table[T[n], {n, 1, 30}] // Flatten (* Jean-François Alcover, May 22 2015, after Alois P. Heinz *)

Table[Length[Select[IntegerPartitions[n], OddQ[Times@@#]&&Length[Union[#]]==k&]], {n, 1, 12}, {k, 1, Floor[Sqrt[n]]}] (* Gus Wiseman, Jun 24 2025 *)

CROSSREFS

Row sums are A000009, strict case of A000041.

Row lengths are A000196.

Leading terms are A001227.

A007690 counts partitions with no singletons, complement A183558.

A034296 counts flat partitions, ranks A066311 or A073491.

A047993 counts partitions with max part = length.

A152140 counts partitions into odd parts by length.

A268193 counts partitions by number of maximal anti-runs, strict A384905.

A384881 counts partitions by number of maximal runs.

Cf. A008284, A038348, A047966, A089259, A325325, A384178, A384886, A384887.