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

Denoted by Delta(n) or Delta_1(n) in Glaisher 1907. - Michael Somos, May 17 2013

A069289(n) <= a(n). - Reinhard Zumkeller, Apr 05 2015

A000203, A001227 and this sequence have the same parity: A053866. - Omar E. Pol, May 14 2016

For the g.f.s given below by Somos Oct 29 2005, Jovovic, Oct 11 2002 and Arndt, Nov 09 2010, see the Hardy-Wright reference, proof of Theorem 382, p. 312, with x^2 replaced by x. - Wolfdieter Lang, Dec 11 2016

a(n) is also the total number of parts in all partitions of n into an odd number of equal parts. - Omar E. Pol, Jun 04 2017

It seems that a(n) divides A000203(n) for every n. - Ivan N. Ianakiev, Nov 25 2017 [Yes, see the formula dated Dec 14 2017].

Also, alternating row sums of A126988. - Omar E. Pol, Feb 11 2018

Where a(n) shows the number of equivalence classes of Hurwitz quaternions with norm n (equivalence defined by right multiplication with one of the 24 Hurwitz units as in A055672), A046897(n) seems to give the number of equivalence classes of Lipschitz quaternions with norm n (equivalence defined by right multiplication with one of the 8 Lipschitz units). - R. J. Mathar, Aug 03 2025

Regarded as a triangular table, this is another version of the number of partitions of n into k parts, A008284. - Franklin T. Adams-Watters, Dec 18 2006

From Gus Wiseman, Feb 10 2021: (Start)

T(n,k) is also the number of partitions of n with greatest part k, if we assume the greatest part of an empty partition to be 0. Row n = 9 counts the following partitions:

111111111 22221 333 432 54 63 72 81 9

222111 3222 441 522 621 711

2211111 3321 4221 531 6111

21111111 32211 4311 5211

33111 42111 51111

321111 411111

3111111

(End)

abs(a(n)) = (1/2) * (number of pairs (i,j) satisfying n = i^2 - j^2 and -n <= i,j <= n). - Benoit Cloitre, Jun 14 2003

As A001227(n) is the number of ways to write n as the difference of 3-gonal numbers, a(n) describes the number of ways to write n as the difference of e-gonal numbers for e in {0,1,4,8}. If pe(n):=(1/2)*n*((e-2)*n+(4-e)) is the n-th e-gonal number, then 4*a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e=1, 2*a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e in {0,4} and for a(n) itself is a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e=8. (Same for e=-1 see A035218.) - Volker Schmitt (clamsi(AT)gmx.net), Nov 09 2004

An argument by Gareth McCaughan suggests that the average of this sequence is log(2). - Hans Havermann, Feb 10 2013 [Supported by a graph. - Vaclav Kotesovec, Mar 01 2023]

From Keith F. Lynch, Jan 20 2024: (Start)

a(n) takes every possible integer value, positive, negative, and zero. Proof: For all nonnegative integers k, a(3^k) = 1+k, a(2^k) = 1-k.

a(n) takes every possible integer value except 1 and -1 infinitely many times. Proof: a(o^(k-1)) = k and a(4*o^(k-1)) = -k for all positive integers k and odd primes o, of which there are infinitely many. a(n) = 0 iff n = 2 (mod 4). a(n) = 1 iff n = 1. a(n) = -1 iff n = 4.

a(n) takes prime value p only for n = o^(p-1), where o is any odd prime.

Terms have a simple pattern that repeats with a period of 4: Positive, zero, positive, negative.

(End)

Inverse Möbius transform of (-1)^(n+1). - Wesley Ivan Hurt, Jun 22 2024

Invert transform of A000009.

From Gus Wiseman, Jul 31 2022: (Start)

Also the number of ways to choose a multiset partition into distinct constant multisets of a multiset of length n that covers an initial interval of positive integers. This interpretation involves only multisets, not sequences. For example, the a(1) = 1 through a(4) = 11 multiset partitions are:

{{1}} {{1,1}} {{1,1,1}} {{1,1,1,1}}

{{1},{2}} {{1},{1,1}} {{1},{1,1,1}}

{{1},{2,2}} {{1,1},{2,2}}

{{2},{1,1}} {{1},{2,2,2}}

{{1},{2},{3}} {{2},{1,1,1}}

{{1},{2},{1,1}}

{{1},{2},{2,2}}

{{1},{2},{3,3}}

{{1},{3},{2,2}}

{{2},{3},{1,1}}

{{1},{2},{3},{4}}

The non-strict version is A055887.

The strongly normal non-strict version is A063834.

The strongly normal version is A270995.

(End)

T(n,k) = T(n+k,-k).

Sum_{k=-floor(n/2)+(n mod 2)..-1} T(n,k) = A108949(n).

Sum_{k=-floor(n/2)+(n mod 2)..0} T(n,k) = A171966(n).

Sum_{k=1..n} T(n,k) = A108950(n).

Sum_{k=0..n} T(n,k) = A130780(n).

Sum_{k=-1..1} T(n,k) = A239835(n).

Sum_{k<>0} T(n,k) = A171967(n).

T(n,-1) + T(n,1) = A239833(n).

Sum_{k=-floor(n/2)+(n mod 2)..n} k * T(n,k) = A209423(n).

Sum_{k=-floor(n/2)+(n mod 2)..n} (-1)^k*T(n,k) = A081362(n) = (-1)^n*A000700(n).

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

McKay-Thompson series of class 24J for the Monster group.

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n^2, g(n) = -1. - Seiichi Manyama, Nov 15 2017