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

%I A321440 #48 May 10 2025 10:45:34
%S A321440 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,
%T A321440 15,6,12,13,11,15,14,8,10,19,10,12,18,8,16,19,9,12,17,14,16,16,10,15,
%U A321440 21,15,14,20,7,16,25,7,20,21,14,18,18,14,12,26,16,17
%N A321440 Number of partitions of n into consecutive parts, all singletons except the largest.
%C A321440 Number of representations of n as the difference of two distinct triangular numbers, plus any multiple of the order of the larger triangular number.
%C A321440 From _Jeremy Lovejoy_, Nov 10 2022: (Start)
%C A321440 For n > 0, a(n) is also equal to the Hurwitz class number H(8n-1).
%C A321440 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)
%H A321440 Chai Wah Wu, <a href="/A321440/b321440.txt">Table of n, a(n) for n = 0..10000</a>
%H A321440 C. Alfes, K. Bringmann, and J. Lovejoy, <a href="https://doi.org/10.1017/S0305004111000375">Automorphic properties of generating functions for generalized odd rank moments and odd Durfee symbols</a>, Math. Proc. Cambridge Philos. Soc. 151 (2011), no. 3, 385-406.
%H A321440 Dandan Chen and Rong Chen, <a href="https://arxiv.org/abs/2107.04809">Generating Functions of the Hurwitz Class Numbers Associated with Certain Mock Theta Functions</a>, arXiv:2107.04809 [math.NT], 2021.
%H A321440 N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, <a href="https://vimeo.com/314786942">Part I</a>, <a href="https://vimeo.com/314790822">Part 2</a>, <a href="https://oeis.org/A320487/a320487.pdf">Slides.</a> (Mentions this sequence)
%F A321440 From _Jeremy Lovejoy_, Nov 10 2022: (Start)
%F A321440 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)).
%F A321440 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)
%e A321440 Here are the derivations of the terms given. Partitions are listed as strings of digits.
%e A321440 n = 0: (empty partition)
%e A321440 n = 1: 1
%e A321440 n = 2: 11, 2
%e A321440 n = 3: 111, 12, 3
%e A321440 n = 4: 1111, 22, 4
%e A321440 n = 5: 11111, 122, 23, 5
%e A321440 n = 6: 111111, 123, 222, 33, 6
%e A321440 n = 7: 1111111, 1222, 34, 7
%e A321440 n = 8: 11111111, 2222, 233, 44, 8
%e A321440 n = 9: 111111111, 12222, 1233, 234, 333, 45, 9
%e A321440 n = 10: 1111111111, 1234, 22222, 55, (10)
%o A321440 (Python)
%o A321440 from sympy.utilities.iterables import partitions
%o A321440 def A321440(n):
%o A321440     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
%o A321440 (Python)
%o A321440 from __future__ import division
%o A321440 def A321440(n): # a faster program based on the characterization in the comments
%o A321440     if n == 0:
%o A321440         return 1
%o A321440     c = 0
%o A321440     for i in range(n):
%o A321440         mi = i*(i+1)//2 + n
%o A321440         for j in range(i+1,n+1):
%o A321440             k = mi - j*(j+1)//2
%o A321440             if k < 0:
%o A321440                 break
%o A321440             if not k % j:
%o A321440                 c += 1
%o A321440     return c # _Chai Wah Wu_, Nov 09 2018
%Y A321440 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.
%Y A321440 Cf. A259825, A321441, A321443.
%K A321440 nonn
%O A321440 0,3
%A A321440 _Allan C. Wechsler_, Nov 09 2018
%E A321440 More terms from _Chai Wah Wu_, Nov 09 2018