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A087135 Number of numbers m such that A073642(m) = n. 4

%I

%S 1,2,2,4,4,6,8,10,12,16,20,24,30,36,44,54,64,76,92,108,128,152,178,

%T 208,244,284,330,384,444,512,592,680,780,896,1024,1170,1336,1520,1728,

%U 1964,2226,2520,2852,3220,3632,4096,4608,5180,5820,6528,7316,8194,9164,10240,11436,12756,14216,15834

%N Number of numbers m such that A073642(m) = n.

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

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

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

%e n=6: numbers m such that A073642(m)=6: {14,15,20,21,34,35,64,65}, therefore a(6)=8.

%e From _Joerg Arndt_, May 23 2013: (Start)

%e There are a(10-1)=15 partitions of 10 where all parts except possibly the two smallest are distinct:

%e 01: [ 1 1 2 6 ]

%e 02: [ 1 1 3 5 ]

%e 03: [ 1 1 8 ]

%e 04: [ 1 2 3 4 ]

%e 05: [ 1 2 7 ]

%e 06: [ 1 3 6 ]

%e 07: [ 1 4 5 ]

%e 08: [ 1 9 ]

%e 09: [ 2 2 6 ]

%e 10: [ 2 3 5 ]

%e 11: [ 2 8 ]

%e 12: [ 3 3 4 ]

%e 13: [ 3 7 ]

%e 14: [ 4 6 ]

%e 15: [ 5 5 ]

%e 16: [ 10 ]

%e (End)

%p 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 (QPochhammer[-1, x] - 1 + O[x]^58)[[3]] (* _Vladimir Reshetnikov_, Nov 20 2015 *)

%o (PARI) /* From the formula given by _Joerg Arndt_: */

%o {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 for(n=0,60,print1(a(n),", ")) /* _Paul D. Hanna_, Feb 19 2012 */

%o (PARI) {a(n)=polcoeff(sum(m=0,n,x^m*prod(k=0,m-1,1+x^k +x*O(x^n))),n)}

%o for(n=0,60,print1(a(n),", ")) /* _Paul D. Hanna_, Feb 19 2012 */

%Y Cf. A087136.

%K nonn

%O 0,2

%A _Reinhard Zumkeller_, Aug 17 2003

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Last modified March 26 16:33 EDT 2019. Contains 321510 sequences. (Running on oeis4.)