A321440 Number of partitions of n into consecutive parts, all singletons except the largest.

1, 1, 2, 3, 3, 4, 5, 4, 5, 7, 5, 6, 8, 5, 8, 10, 5, 8, 10, 7, 10, 11, 7, 8, 13, 9, 9, 14, 7, 12, 15, 6, 12, 13, 11, 15, 14, 8, 10, 19, 10, 12, 18, 8, 16, 19, 9, 12, 17, 14, 16, 16, 10, 15, 21, 15, 14, 20, 7, 16, 25, 7, 20, 21, 14, 18, 18, 14, 12, 26, 16, 17

Offset: 0

Author

Allan C. Wechsler, Nov 09 2018

Keywords

• nonn

Comments

Number of representations of n as the difference of two distinct triangular numbers, plus any multiple of the order of the larger triangular number.

From Jeremy Lovejoy, Nov 10 2022: (Start)

For n > 0, a(n) is also equal to the Hurwitz class number H(8n-1).

a(n) is also equal to the number of partitions y of n having no repeated even parts and smallest part odd, counted according to the weight w(y) = (-1)^(the number of even parts)*(the number of occurrences of the smallest part). For example, the partitions of 6 having no repeated even parts and smallest part odd are [5,1], [4,1,1], [3,3], [3,2,1], [3,1,1,1], [2,1,1,1,1], and [1,1,1,1,1,1], which are counted with weights 1,-2,2,-1,3,-4, and 6, giving a(6) = 1-2+2-1+3-4+6 = 5. (End)

Examples

			Here are the derivations of the terms given. Partitions are listed as strings of digits.
n = 0: (empty partition)
n = 1: 1
n = 2: 11, 2
n = 3: 111, 12, 3
n = 4: 1111, 22, 4
n = 5: 11111, 122, 23, 5
n = 6: 111111, 123, 222, 33, 6
n = 7: 1111111, 1222, 34, 7
n = 8: 11111111, 2222, 233, 44, 8
n = 9: 111111111, 12222, 1233, 234, 333, 45, 9
n = 10: 1111111111, 1234, 22222, 55, (10)
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..10000
• C. Alfes, K. Bringmann, and J. Lovejoy, Automorphic properties of generating functions for generalized odd rank moments and odd Durfee symbols, Math. Proc. Cambridge Philos. Soc. 151 (2011), no. 3, 385-406.
• Dandan Chen and Rong Chen, Generating Functions of the Hurwitz Class Numbers Associated with Certain Mock Theta Functions, arXiv:2107.04809 [math.NT], 2021.
• N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)

Crossrefs

See comment by Emeric Deutsch at A001227 (partitions into consecutive parts, all singletons); the partitions considered in the present sequence are a superset of those described by Deutsch.

Cf. A259825, A321441, A321443.

Programs

Python

from sympy.utilities.iterables import partitions
def A321440(n):
    return 1 if n == 0 else sum(1 for s,p in partitions(n,size=True) if len(p)-1 == max(p)-min(p) == s-p[max(p)]) # Chai Wah Wu, Nov 09 2018

Python

from _future_ import division
def A321440(n): # a faster program based on the characterization in the comments
    if n == 0:
        return 1
    c = 0
    for i in range(n):
        mi = i*(i+1)//2 + n
        for j in range(i+1,n+1):
            k = mi - j*(j+1)//2
            if k < 0:
                break
            if not k % j:
                c += 1
    return c # Chai Wah Wu, Nov 09 2018

Formula

From Jeremy Lovejoy, Nov 10 2022: (Start)

G.f.: 1 + Sum_{n>=0} x^(n+1)*Product_{k=1..n} (1-x^(2*k))/Product_{k=1..n+1} (1-x^(2*k-1)).

G.f.: 1 + Sum_{n>=1} (-1)^(n+1)*x^(n^2)/((1-x^(2*n-1))*Product_{k=1..n} (1-x^(2*k-1))). (End)

Extensions

More terms from Chai Wah Wu, Nov 09 2018