cp's OEIS Frontend

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

%I A087135 #28 Aug 25 2019 10:41:19
%S A087135 1,2,2,4,4,6,8,10,12,16,20,24,30,36,44,54,64,76,92,108,128,152,178,
%T A087135 208,244,284,330,384,444,512,592,680,780,896,1024,1170,1336,1520,1728,
%U A087135 1964,2226,2520,2852,3220,3632,4096,4608,5180,5820,6528,7316,8194,9164,10240,11436,12756,14216,15834
%N A087135 Number of positive numbers m such that A073642(m) = n.
%C A087135 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
%C A087135 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
%F A087135 a(n) = 2*A000009(n) for n>0.
%F A087135 G.f.: Sum_{n>=0} (x^(n*(n+1)/2) / Product_{k=1..n+1} (1-x^k ) ). - _Joerg Arndt_, Mar 24 2011
%F A087135 G.f.: Sum_{n>=0} x^n * Product_{k=0..n-1} (1+x^k). - _Paul D. Hanna_, Feb 19 2012
%e A087135 n=6: numbers m such that A073642(m)=6: {14,15,20,21,34,35,64,65}, therefore a(6)=8.
%e A087135 From _Joerg Arndt_, May 23 2013: (Start)
%e A087135 There are a(10-1)=15 partitions of 10 where all parts except possibly the two smallest are distinct:
%e A087135 01:  [ 1 1 2 6 ]
%e A087135 02:  [ 1 1 3 5 ]
%e A087135 03:  [ 1 1 8 ]
%e A087135 04:  [ 1 2 3 4 ]
%e A087135 05:  [ 1 2 7 ]
%e A087135 06:  [ 1 3 6 ]
%e A087135 07:  [ 1 4 5 ]
%e A087135 08:  [ 1 9 ]
%e A087135 09:  [ 2 2 6 ]
%e A087135 10:  [ 2 3 5 ]
%e A087135 11:  [ 2 8 ]
%e A087135 12:  [ 3 3 4 ]
%e A087135 13:  [ 3 7 ]
%e A087135 14:  [ 4 6 ]
%e A087135 15:  [ 5 5 ]
%e A087135 16:  [ 10 ]
%e A087135 (End)
%p A087135 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
%t A087135 (QPochhammer[-1, x] - 1 + O[x]^58)[[3]] (* _Vladimir Reshetnikov_, Nov 20 2015 *)
%o A087135 (PARI) /* From the formula given by _Joerg Arndt_: */
%o A087135 {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)}
%o A087135 for(n=0,60,print1(a(n),", ")) /* _Paul D. Hanna_, Feb 19 2012 */
%o A087135 (PARI) {a(n)=polcoeff(sum(m=0,n,x^m*prod(k=0,m-1,1+x^k +x*O(x^n))),n)}
%o A087135 for(n=0,60,print1(a(n),", ")) /* _Paul D. Hanna_, Feb 19 2012 */
%Y A087135 Cf. A087136.
%K A087135 nonn
%O A087135 0,2
%A A087135 _Reinhard Zumkeller_, Aug 17 2003
%E A087135 Added "positive" to definition. - _N. J. A. Sloane_, Aug 25 2019