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
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
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Aug 17 2003
EXTENSIONS
Added "positive" to definition. - N. J. A. Sloane, Aug 25 2019
STATUS
approved