

A049988


Number of nondecreasing arithmetic progressions of positive integers with sum n.


66



1, 1, 2, 3, 4, 4, 7, 5, 7, 9, 9, 7, 14, 8, 11, 16, 13, 10, 20, 11, 17, 21, 16, 13, 27, 17, 18, 26, 22, 16, 35, 17, 23, 31, 23, 25, 41, 20, 25, 36, 33, 22, 46, 23, 31, 48, 30, 25, 52, 29, 38, 47, 36, 28, 57, 37, 41, 52, 37, 31, 71, 32, 39, 62, 44, 43, 69, 35, 45, 62, 57, 37, 79, 38
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OFFSET

0,3


COMMENTS

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)
We show how Leroy Quet's g.f. Sum_{n >= 0} a(n)*x^n = 1/(1x) + Sum_{k >= 2} x^k/(1x^(k*(k1)/2))/(1x^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(k1)  x^k + 1) = Sum_{k >= 2} x^t(k)/((1  x^k) * (1  x^t(k1))), 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



FORMULA

G.f.: 1/(1x) + Sum_{k>=2} x^k/(1x^(k*(k1)/2))/(1x^k).  Leroy Quet, Apr 08 2010. [Edited by Gus Wiseman, May 03 2019]


MATHEMATICA

a[n_]:=If[n==0, 1, Block[{i, c=Floor[(n1)/2]+DivisorSigma[0, n]}, Do[i=1; While[i*k<n, If[Mod[2*(ni*k), k*(k1)]==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/(1x) + sum(k=2, n, x^k/(1  x^(k*(k1)/2))/(1x^k) + O(x*x^n)))} \\ Andrew Howroyd, Sep 28 2019


CROSSREFS

Cf. A000005, A000217, A007862, A047966, A049982, A049983, A049986, A049987, A129654, A240026, A240027, A307824, A320466, A325325, A325328.


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



