0,5

Row sums = A000041, the partition numbers.

The current triangle is the 2nd in an infinite set, followed by A174714 (k=3), and A174715, (k=4); in which row sums of each triangle = A000041.

k-th triangle in the infinite set can be defined as having the sequence:

"Euler transform of ones: (1,1,1,...) interleaved with (k-1) zeros"; shifted down k times (except column 0) in successive columns, then multiplied * triangle A174712, the diagonalized variant of A000041, A174713 begins with A000009 shifted down twice (triangle A173305); where A000009 = the Euler transform of period 2 sequence: [1,0,1,0,...].

Similarly, triangle A174714 begins with A000716 shifted down thrice; where A000716 = the Euler transform of period 3 series: [1,1,0,1,1,0,...]. Then multiply the latter as an infinite lower triangular matrix * A174712, the diagonalized variant of A000041, obtaining triangle A174714 with row sums = A000041.

Case k=4 = triangle A174715 which begins with the Euler transform of period 4 series: [1,1,1,0,1,1,1,0,...], shifted down 4 times in successive columns then multiplied * A174712, the diagonalized variant of A000041.

All triangles in the infinite set have row sums = A000041.

The sequences: "Euler transform of ones interleaved with (k-1) zeros" have the following properties, beginning with k=2:

...

k=2, A000009: = Euler transform of [1,0,1,0,1,0,...] and satisfies

.....A000009. = p(x)/p(x^2), where p(x) = polcoeff A000041; and A000041 =

.....A000009(x) = r(x), then p(x) = r(x) * r(x^2) * r(x^4) * r(x^8) * ...

...

k=3, A000726: = Euler transform of [1,1,0,1,1,0,...] and satisfies

.....A000726(x): = p(x)/p(x^3), and given s(x) = polcoeff A000726, we get

.....A000041(x) = p(x) = s(x) * s(x^3) * s(x^9) * s(x^27) * ...

...

k=4, A001935: = Euler transform of [1,1,1,0,1,1,1,0,...] and satisfies

.....A001935(x) = p(x)/p(x^4) and given t(x) = polcoeff A001935, we get

.....A000041(x) = p(x) = t(x) * t(x^4) * t(x^16) * t(x^64) * ...

...

Also the number of integer partitions of n whose even parts sum to k, for k an even number from zero to n. The version including odd k is A113686. - Gus Wiseman, Oct 23 2023

As infinite lower triangular matrices, A173305 * A174712.

T(n,k) = A000009(n-2k) * A000041(k). - Gus Wiseman, Oct 23 2023

First few rows of the triangle =

1;

1;

1, 1;

2, 1;

2, 1, 2;

3, 2, 2;

4, 2, 2, 3;

5, 3, 4, 3;

6, 4, 4, 3, 5;

8, 5, 6, 6, 5;

10, 6, 8, 6, 5, 7;

12, 8, 10, 9, 10, 7;

15, 10, 12, 12, 10, 7, 11;

18, 12, 16, 15, 15, 14, 11;

22, 15, 20, 18, 20, 14, 11, 15;

...

From Gus Wiseman, Oct 23 2023: (Start)

Row n = 9 counts the following partitions:

(9) (72) (54) (63) (81)

(711) (5211) (522) (6111) (621)

(531) (3321) (4311) (432) (441)

(51111) (321111) (411111) (42111) (4221)

(333) (21111111) (32211) (3222) (22221)

(33111) (2211111) (222111)

(3111111)

(111111111)

(End)

Table[Length[Select[IntegerPartitions[n], Total[Select[#, EvenQ]]==k&]], {n, 0, 15}, {k, 0, n, 2}] (* Gus Wiseman, Oct 23 2023 *)

Cf. A000009, A173305, A174712, A174714, A174715.

Row sums are A000041.

The odd version is A365067.

The corresponding rank statistic is A366531, odd version A366528.

A000009 counts partitions into odd parts, ranks A066208.

A113685 counts partitions by sum of odd parts, even version A113686.

A239261 counts partitions with (sum of odd parts) = (sum of even parts).

Cf. A035363, A066967, A130780, A171966, A182616, A366527, A366533.

Sequence in context: A060272 A209315 A352218 * A129985 A085243 A381596

Adjacent sequences: A174710 A174711 A174712 * A174714 A174715 A174716

nonn,tabf

Gary W. Adamson, Mar 27 2010

approved