cp's OEIS Frontend

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

Coefficients of replicable function number 96a. - N. J. A. Sloane, Jun 10 2015

For n >= 1, a(n) is the minimal row sum in the character table of the symmetric group S_n. The minimal row sum in the table corresponds to the one-dimensional alternating representation of S_n. The maximal row sum is in sequence A085547. - Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 15 2003

Also the number of partitions of n into parts != 2 and differing by >= 6 with strict inequality if a part is even. [Alladi]

Let S be the set formed by the partial sums of 1+[2,3]+[2,5]+[2,7]+[2,9]+..., where [2,odd] indicates a choice, e.g., we may have 1+2, or 1+3+2, or 1+3+5+2+9, etc. Then A000700(n) is the number of elements of S that equal n. Also A000700(n) is the same parity as A000041(n) (the partition numbers). - Jon Perry, Dec 18 2003

a(n) is for n >= 2 the number of conjugacy classes of the symmetric group S_n which split into two classes under restriction to A_n, the alternating group. See the G. James - A. Kerber reference given under A115200, p. 12, 1.2.10 Lemma and the W. Lang link under A115198.

Also number of partitions of n such that if k is the largest part, then k occurs an odd number of times and each integer from 1 to k-1 occurs a positive even number of times (these are the conjugates of the partitions of n into distinct odd parts). Example: a(15)=4 because we have [3,3,3,2,2,1,1], [3,2,2,2,2,1,1,1,1], [3,2,2,1,1,1,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Apr 16 2006

The INVERTi transform of A000009 (number of partitions of n into odd parts starting with offset 1) = (1, 0, 1, -1, 1, -1, 1, -2, 2, -2, 2, -3, 3, -3, 4, ...); = left border of triangle A146061. - Gary W. Adamson, Oct 26 2008

For n even: the sum over all even nonnegative integers, k, such that k^2 < n, of the number of partitions of (n-k^2)/2 into parts of size at most k. For n odd: the sum over all odd nonnegative integers, j, such that j^2 < n, of the number of partitions of (n-j^2)/2 into parts of size at most j. - Graham H. Hawkes, Oct 18 2013

This number is also (the number of conjugacy classes of S_n containing even permutations) - (the number of conjugacy classes of S_n containing odd permutations) = (the number of partitions of n into a number of parts having the same parity as n) - (the number of partitions of n into a number of parts having opposite parity as n) = (the number of partitions of n with largest part having same parity as n) - (the number of partitions with largest part having opposite parity as n). - David L. Harden, Dec 09 2016

a(n) is odd iff n belongs to A052002; that is, Sum_{n>=0} x^A052002(n) == Sum_{n>=0} a(n)*x^n (mod 2). - Peter Bala, Jan 22 2017

Also the number of conjugacy classes of S_n whose members yield unique square roots, i.e., there exists a unique h in S_n such that hh = g for any g in such a conjugacy class. Proof: first note that a permutation's square roots are determined by the product of the square roots of its decomposition into cycles of different lengths. h can only travel to one other cycle before it must "return home" (h^2(x) = g(x) must be in x's cycle), and, because if g^n(x) = x then h^2n(x) = x and h^2n(h(x)) = h(x), this "traveling" must preserve cycle length or one cycle will outpace the other. However, a permutation decomposing into two cycles of the same length has multiple square roots: for example, e = e^2 = (a b)^2, (a b)(c d) = (a c b d)^2 = (a d b c)^2, (a b c)(d e f) = (a d b e c f)^2 = (a e b f c d)^2, etc. This is true for any cycle length so we need only consider permutations with distinct cycle lengths. Finally, even cycle lengths are odd permutations and thus cannot be square, while odd cycle lengths have the unique square root h(x) = g^((n+1)/2)(x). Thus there is a correspondence between these conjugacy classes and partitions into distinct odd parts. - Keith J. Bauer, Jan 09 2024

a(2*n) equals the number of partitions of n into parts congruent to +-2, +-3, +-4 or +-5 mod 16. See Merca, 2015, Corollary 4.3. - Peter Bala, Dec 12 2024

The main array with 0 and 1 interchanged is A115198.

A partition of n is (here) called even, resp. odd, if the number of even parts is even, resp. odd. A partition with no (0) even part is therefore even.

The row length sequence of this triangle is p(n)=A000041(n) (number of partitions).

See the W. Lang link under A115198 for the first 10 rows where 0 and 1 should be swapped for this a(n,m) entry.

A conjugacy class of the symmetric group S_n with the cycle structure given by the partition, listed in the A-St order, consists of even, resp. odd, permutations if a(n,m) is even, resp. odd.

See A115198 for the parity of a(n,m) with 1 for even, 0 for odd (main entry).

See A115199 for the parity of a(n,m) with 0 for even, 1 for odd.

The parity of these numbers determines whether a conjugacy class of the symmetric group S_n, which is determined by its cycle structure, consists of even or odd permutations.

The row length sequence of this triangle is p(n)=A000041(n) (number of partitions).