cp's OEIS Frontend

Previous name was: Invert transform of right-shifted partition function (A000041).

Sums of the antidiagonals of the array formed by sequences A000007, A000041, A000712, A000716, ... or its transpose A000012, A000027, A000096, A006503, A006504, ....

Row sums of triangle A143866 = (1, 2, 5, 12, 29, 69, 165, ...) and right border of A143866 = (1, 1, 2, 5, 12, ...). - Gary W. Adamson, Sep 04 2008

Starting with offset 1 = A137682 / A000041; i.e. (1, 3, 7, 17, 40, 96, ...) / (1, 2, 3, 5, 7, 11, ...). - Gary W. Adamson, May 01 2009

From L. Edson Jeffery, Mar 16 2011: (Start)

Another approach is the following. Let T be the infinite lower triangular matrix with columns C_k (k=0,1,2,...) such that C_0=A000041 and, for k > 0, such that C_k is the sequence giving the number of partitions of n into parts of k+1 kinds (successive self-convolutions of A000041 yielding A000712, A000716, ...) and shifted down by k rows. Then T begins (ignoring trailing zero entries in the rows)

(1, 0, ... )

(1, 1, 0, ... )

(2, 2, 1, 0, ... )

(3, 5, 3, 1, 0, ... )

(5, 10, 9, 4, 1, 0, ...)

etc., and a(n) is the sum of entries in row n of T. (End)

Number of partitions of n into an even number of distinct parts minus number of partitions of n into an odd number of distinct parts, with 2 types of each part. E.g., for n=4, we consider k and k* to be different versions of k and so we have 4, 4*, 31, 31*, 3*1, 3*1*, 22*, 211*, 2*11*. The even partitions number 5 and the odd partitions number 4, so a(4)=5-4=1. - Jon Perry, Apr 04 2004

Also, number of partitions of n into parts of -2 different kinds (based upon formal analogy). - Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 29 2004

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

Number 68 of the 74 eta-quotients listed in Table I of Martin (1996).

Equals row sums of triangle A147767. - Gary W. Adamson, Nov 11 2008

Also number of partitions of n into parts of 2 kinds except for 1. - Reinhard Zumkeller, Nov 06 2012

Antidiagonal sums of triangle A093010.

Suppose that p, with parts x(1) >= x(2) >= ... >= x(k), is a partition of n. Define AS(p), the alternating sum of p, by x(1) - x(2) + x(3) - ... + ((-1)^(k-1))*x(k); note that AS(p) has the same parity as n. Column 1 is given by T(n,1) = (number of partitions of 2n-1 having AS(p) = 1) = A000070(n) for n >= 1. Columns 2 and 3 are essentially A000098 and A103924, and the limiting column (after deleting initial 0's), A000712. The sum of numbers in row n is A000041(2n-1). The corresponding array for partitions into distinct parts is given by A152157 (defined as the number of partitions of 2n+1 into 2k+1 odd parts).

The folds are always made so the longer side becomes the shorter side.

We could have counted not only the holes but also all the notches: 4, 6, 9, 15, 25, 45, 81, 153, 289, ... which has the formula a(n) = (2^ceiling(n/2) + 1) * (2^floor(n/2) + 1) and appears to match the sequence A183978. - Philippe Gibone, Jul 06 2016

The same sequence (0,0,1,3,9,21,49,...) turns up when you start with an isosceles right triangular piece of paper and repeatedly fold it in half, snipping corners as you go. Is there an easy way to see why the two questions have the same answer? - James Propp, Jul 05 2016

Reply from Tom Karzes, Jul 05 2016: (Start)

This case seems a little more complicated than the rectangular case, since with the triangle you alternate between horizontal/vertical folds vs. diagonal folds, and the resulting fold pattern is more complex, but I think the basic argument is essentially the same.

Note that with the triangle, the first hole doesn't appear until after you've made 3 folds, so if you start counting at zero folds, you have three leading zeros in the sequence: 0,0,0,1,3,9,21,... (End)

Also the number of subsets of {1,2,...,n} that contain both even and odd numbers. For example, a(3)=3 and the 3 subsets are {1,2}, {2,3}, {1,2,3}; a(4)=9 and the 9 subsets are {1,2}, {1,4}, {2,3}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}, {1,2,3,4}. (See comments in A052551 for the number of subsets of {1,2,...,n} that contain only odd and even numbers.) - Enrique Navarrete, Mar 26 2018

Also the number of integer compositions of n + 1 with an odd part other than the first or last. The complementary compositions are counted by A052955(n>0) = A027383(n) + 1. - Gus Wiseman, Feb 05 2022

Also the number of unit squares in the (n+1)-st iteration in the version of the dragon curve where the rotation directions alternate, so that any clockwise rotation is followed by a counterclockwise rotation, and vice versa (see image link below). - Talmon Silver, May 09 2023

a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018

A subset-product of an integer partition y is a product of some submultiset of y. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Number of distinct values obtained when A003963 is applied to all divisors of n. - Antti Karttunen, Sep 05 2018

Also the first (k+1)/2 terms of this sequence are the levels of the (k+1)/2 plateaus of the column k of A195825, whose lengths are k+1, k-1, k-3, k-5,... 2, if k is odd.

Also the first k/2 terms of this sequence are the levels of the k/2 plateaus of the column k of A195825, whose lengths are k+1, k-1, k-3, k-5,... 3, if k is a positive even number.

For the visualization of the plateaus see the graph of the sequences mentioned in crossrefs section (columns k=1..10 of A195825), for example see the graph of A210964.

Also numbers that are repeated in column k of square array A195825, when k -> infinity.

Note that the definition and the comments related to the square array A195825 mentioned above are also valid for the square array A211970, since both arrays contains the same columns, if k >= 1.

Is this the EULER transform of 4, 3, 3, 3, 3, 3, 3...?

Number of proper partitions of n into parts of two kinds (i.e. both kinds must be present). - Franklin T. Adams-Watters, Feb 08 2006