cp's OEIS Frontend

Also number of partitions of n where no part appears more than three times.

a(n) satisfies Euler's pentagonal number (A001318) theorem, unless n is in A062717 (see Fink et al.).

Also number of partitions of n in which the least part and the differences between consecutive parts is at most 3. Example: a(5)=6 because we have [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1] and [1,1,1,1,1]. - Emeric Deutsch, Apr 19 2006

Equals A000009 convolved with its aerated variant, = polcoeff A000009 * A000041 * A010054 (with alternate signs). - Gary W. Adamson, Mar 16 2010

Equals left border of triangle A174715. - Gary W. Adamson, Mar 27 2010

The Cayley reference is actually to A083365. - Michael Somos, Feb 24 2011

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Convolution of A000009 and A035457. - Vaclav Kotesovec, Aug 23 2015

Convolution inverse is A082303. - Michael Somos, Sep 30 2017

The g.f. in the form Sum_{n >= 0} x^(n*(n+1)/2) * Product_{k = 1..n} (1+x^k)/(1-x^k) = Sum_{n >= 0} x^(n*(n+1)/2) * Product_{k = 1..n} (1+x^k)/(1+x^k-2*x^k) == Sum_{n >= 0} x^(n*(n+1)/2) (mod 2). It follows that a(n) is odd iff n = k*(k + 1)/2 for some nonnegative integer k. Cf. A333374. - Peter Bala, Jan 08 2025

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

Eigensequence of the triangle = A058694: (1, 1, 2, 6, 30, 210,...), i.e., given A058694 preceded by a "1", triangle A174712 * the latter variant = the same sequence but shifted left.

Refer to comments in A174713.

Row sums = A000041, the partition numbers.