OFFSET
0,3
COMMENTS
From Gus Wiseman, May 03 2019: (Start)
a(n) is the number of integer partitions of n with equal differences. The Heinz numbers of these partitions are given by A325328. For example, the a(1) = 1 through a(9) = 9 partitions are:
1 2 3 4 5 6 7 8 9
11 21 22 32 33 43 44 54
111 31 41 42 52 53 63
1111 11111 51 61 62 72
222 1111111 71 81
321 2222 333
111111 11111111 432
531
111111111
(End)
From Petros Hadjicostas, Sep 29 2019: (Start)
We show how Leroy Quet's g.f. Sum_{n >= 0} a(n)*x^n = 1/(1-x) + Sum_{k >= 2} x^k/(1-x^(k*(k-1)/2))/(1-x^k) in the Formula section below can be derived from Graeme McRae's g.f. for A049982 (see one of the links below).
Let b(n) = A049982(n) for n >= 1. Then Graeme McRae proved that Sum_{n >= 1} b(n)*x^n = Sum_{k >= 2} x^t(k)/(x^t(k) - x^t(k-1) - x^k + 1) = Sum_{k >= 2} x^t(k)/((1 - x^k) * (1 - x^t(k-1))), where t(k) = A000217(k) = k*(k+1)/2.
Since a(n) - b(n) = A000005(n) for n >= 1, to finish the proof, we only need to show that K(x) := 1 + Sum_{n >= 1} a(n)*x^n - Sum_{n >= 1} b(n)*x^n is the g.f. of A000005 (= number of divisors). But it is easy to show that K(x) = 1 + Sum_{k >= 1} x^k/(1 - x^k) = 1 + Sum_{n >= 1} A000005(n)*x^n (Lambert series for the number of divisors function). (End)
LINKS
Lars Blomberg, Table of n, a(n) for n = 0..10000 (Corrected by Gus Wiseman, May 03 2019)
Lars Blomberg, C# program for calculating b-file (needs to be updated for a(0) = 1 - Gus Wiseman, May 07 2019).
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions with an odd common difference, Integers 9(1) (2009), 77-81.
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
F. Javier de Vega, A Complete Solution of the Partitions of a Number into Arithmetic Progressions, arXiv:2004.09505 [math.NT], 2020.
F. Javier de Vega, On the parabolic partitions of a number, J. Alg., Num. Theor., and Appl. (2023) Vol. 61, No. 2, 135-169.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
Augustine O. Munagi, Combinatorics of integer partitions in arithmetic progression, Integers 10(1) (2010), 73-82.
Augustine O. Munagi and Temba Shonhiwa, On the partitions of a number into arithmetic progressions, Journal of Integer Sequences 11 (2008), Article 08.5.4.
Wikipedia, Arithmetic progression.
Wikipedia, Lambert series.
FORMULA
G.f.: 1/(1-x) + Sum_{k>=2} x^k/(1-x^(k*(k-1)/2))/(1-x^k). - Leroy Quet, Apr 08 2010. [Edited by Gus Wiseman, May 03 2019]
a(n) = A049982(n) + A000005(n) = A049980(n) + A000005(n) - 1 for n >= 1. - Petros Hadjicostas, Sep 28 2019
MATHEMATICA
a[n_]:=If[n==0, 1, Block[{i, c=Floor[(n-1)/2]+DivisorSigma[0, n]}, Do[i=1; While[i*k<n, If[Mod[2*(n-i*k), k*(k-1)]==0, c++]; i++], {k, 3, (Sqrt[1+8*n]-1)/2}]; c]]; a/@Range[0, 73] (* Giovanni Resta, Feb 16 2013. Edited by Gus Wiseman, May 07 2019 *)
Table[Length[Select[IntegerPartitions[n], SameQ@@Differences[#]&]], {n, 0, 30}] (* Gus Wiseman, May 03 2019 *)
PROG
(PARI) seq(n)={Vec(1/(1-x) + sum(k=2, n, x^k/(1 - x^(k*(k-1)/2))/(1-x^k) + O(x*x^n)))} \\ Andrew Howroyd, Sep 28 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Edited by Max Alekseyev, May 03 2010
a(0) = 1 prepended by Gus Wiseman, May 03 2019
STATUS
approved