A238005 Number of partitions of n into distinct parts such that (greatest part) - (least part) = (number of parts).

0, 0, 0, 1, 0, 1, 1, 2, 0, 2, 2, 2, 2, 2, 1, 4, 3, 2, 3, 3, 2, 4, 4, 4, 3, 4, 2, 5, 5, 3, 5, 6, 3, 5, 3, 5, 6, 6, 4, 6, 6, 4, 6, 6, 3, 7, 7, 7, 6, 6, 5, 7, 7, 5, 6, 8, 6, 8, 8, 6, 8, 8, 4, 9, 6, 7, 9, 9, 7, 7, 9, 8, 9, 9, 5, 9, 7, 8, 10, 10

Offset

1,8

Comments

Note that partitions into distinct parts are also called strict partitions.

a(n) is the number of strict partitions of n into nearly consecutive parts, that is, the number of ways to write n as a sum of terms i, i+1, i+2, ..., i+k (i>=1, k>=2) where one of the interior parts i+1, i+2, ..., i+k-1 is missing. Examples of nearly consecutive partitions (corresponding to the initial nonzero values of a(n)) are 13, 24, 124, 134, 35, 235, 46, ... . - Don Reble, Sep 07 2021

Let T(n) = n*(n+1)/2 = A000217(n) denote the n-th triangular number.

Theorem A. a(n) = b(n) - c(n), where b(n) is the inverse triangular number sequence A003056, that is, b(n) is the maximal i such that T_i <= n, and c(n) is the number of partitions of n into consecutive parts = number of odd divisors of n = A001227(n).

This theorem was conjectured by Omar E. Pol in February 2018, and proved independently by William J. Keith and Roland Bacher on Sep 05 2021. The elegant proof given in the link below is due to Don Reble.

Links

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

N. J. A. Sloane, Proof of Theorem A in A238005

Formula

G.f. = (x/(1-x)) * Sum_{k >= 1} x^(k*(k+1)/2) * (1 - x^(k-1)) / (1 - x^k). This follows from Theorem A and the g.f.s for A003056 and A001227. - William J. Keith, Sep 05 2021

a(n) = A238007(n) - A238006(n). - Omar E. Pol, Sep 11 2021

A001227(n) + a(n) + A238006(n) = A000009(n). - R. J. Mathar, Sep 23 2021

Example

a(8) = 2 counts these partitions:  53, 431.

Mathematica

z = 70; q[n_] := q[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; p1[p_] := p1[p] = DeleteDuplicates[p]; t[p_] := t[p] = Length[p1[p]];
Table[Count[q[n], p_ /; Max[p] - Min[p] < t[p]], {n, z}]  (* A001227 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] <= t[p]], {n, z}] (* A003056 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] == t[p]], {n, z}] (* A238005 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] > t[p]], {n, z}]  (* A238006 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] >= t[p]], {n, z}] (* A238007 *)
{0}~Join~Array[Floor[(Sqrt[1 + 8 #] - 1)/2] - DivisorSum[#, 1 &, OddQ] &, 102] (* Michael De Vlieger, Feb 18 2018 *)

Prog

(PARI) a(n) = if (n, (sqrtint(8*n+1)-1)\2 - sumdiv(n, d, d%2), 0); \\ Michel Marcus, Mar 01 2018

Crossrefs

Cf. A000009, A000217, A001227, A003056, A238006, A238007.

a(n) is also the number of zeros in the n-th row of the triangles A196020, A211343, A231345, A236106, A237048 (simpler), A239662, A261699, A271344, A272026, A280850, A285574, A285891, A285914, A286013, A296508 (and possibly others). Omar E. Pol, Feb 17 2018

Row sums of A347579. - Omar E. Pol, Sep 07 2021

Sequence in context: A044944 A339432 A044945 * A296509 A089582 A044946

Adjacent sequences: A238002 A238003 A238004 * A238006 A238007 A238008

Keyword

nonn,easy

Author

Clark Kimberling, Feb 17 2014

Extensions

Edited by N. J. A. Sloane, Sep 11 2021, mostly to add Theorem A.

Status

approved