A087135 Number of positive numbers m such that A073642(m) = n.

1, 2, 2, 4, 4, 6, 8, 10, 12, 16, 20, 24, 30, 36, 44, 54, 64, 76, 92, 108, 128, 152, 178, 208, 244, 284, 330, 384, 444, 512, 592, 680, 780, 896, 1024, 1170, 1336, 1520, 1728, 1964, 2226, 2520, 2852, 3220, 3632, 4096, 4608, 5180, 5820, 6528, 7316, 8194, 9164, 10240, 11436, 12756, 14216, 15834

Offset

0,2

Comments

For n > 0, number of partitions of n into distinct nonnegative integers; for all n, number of nonempty partitions of n into distinct nonnegative integers. - Franklin T. Adams-Watters, Dec 28 2006

For n >= 1, a(n-1) is the number of partitions of n where all parts except possibly the two smallest are distinct, see example. - Joerg Arndt, May 23 2013

Links

Table of n, a(n) for n=0..57.

Formula

a(n) = 2*A000009(n) for n>0.

G.f.: Sum_{n>=0} (x^(n*(n+1)/2) / Product_{k=1..n+1} (1-x^k ) ). - Joerg Arndt, Mar 24 2011

G.f.: Sum_{n>=0} x^n * Product_{k=0..n-1} (1+x^k). - Paul D. Hanna, Feb 19 2012

Example

n=6: numbers m such that A073642(m)=6: {14,15,20,21,34,35,64,65}, therefore a(6)=8.
From Joerg Arndt, May 23 2013: (Start)
There are a(10-1)=15 partitions of 10 where all parts except possibly the two smallest are distinct:
01:  [ 1 1 2 6 ]
02:  [ 1 1 3 5 ]
03:  [ 1 1 8 ]
04:  [ 1 2 3 4 ]
05:  [ 1 2 7 ]
06:  [ 1 3 6 ]
07:  [ 1 4 5 ]
08:  [ 1 9 ]
09:  [ 2 2 6 ]
10:  [ 2 3 5 ]
11:  [ 2 8 ]
12:  [ 3 3 4 ]
13:  [ 3 7 ]
14:  [ 4 6 ]
15:  [ 5 5 ]
16:  [ 10 ]
(End)

Maple

ZL:=product(1+x^(j-1), j=1..59): gser:=series(ZL, x=0, 55): seq(coeff(gser, x, n), n=1..48); # Zerinvary Lajos, Mar 09 2007

Mathematica

(QPochhammer[-1, x] - 1 + O[x]^58)[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

Prog

(PARI) /* From the formula given by Joerg Arndt: */
{a(n)=polcoeff(sum(m=0, n, x^(m*(m+1)/2)/prod(k=1, m+1, 1-x^k +x*O(x^n))), n)}
for(n=0, 60, print1(a(n), ", ")) /* Paul D. Hanna, Feb 19 2012 */
(PARI) {a(n)=polcoeff(sum(m=0, n, x^m*prod(k=0, m-1, 1+x^k +x*O(x^n))), n)}
for(n=0, 60, print1(a(n), ", ")) /* Paul D. Hanna, Feb 19 2012 */

Crossrefs

Cf. A087136.

Sequence in context: A211859 A057601 A294150 * A227135 A162417 A240012

Adjacent sequences: A087132 A087133 A087134 * A087136 A087137 A087138

Keyword

nonn

Author

Reinhard Zumkeller, Aug 17 2003

Extensions

Added "positive" to definition. - N. J. A. Sloane, Aug 25 2019

Status

approved