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 A025147 Number of partitions of n into distinct parts >= 2. 38

%I

%S 1,0,1,1,1,2,2,3,3,5,5,7,8,10,12,15,17,21,25,29,35,41,48,56,66,76,89,

%T 103,119,137,159,181,209,239,273,312,356,404,460,522,591,669,757,853,

%U 963,1085,1219,1371,1539,1725,1933,2164,2418,2702,3016,3362,3746,4171,4637,5155

%N Number of partitions of n into distinct parts >= 2.

%C Comment from _R. J. Mathar_, Jul 31 2008: (Start)

%C These "partitions of n into distinct parts >= k" and "partitions of n into distinct parts, the least being k-1" come in pairs of similar, almost shifted but not identical, sequences:

%C A025147, A096765 (k=2)

%C A025148, A096749 (k=3)

%C A025149, A026824 (k=4)

%C A025150, A026825 (k=5)

%C A025151, A026826 (k=6)

%C A025152, A026827 (k=7)

%C A025153, A026828 (k=8)

%C A025154, A026829 (k=9)

%C A025155, A026830 (k=10)

%C A096740, A026831 (k=11)

%C The distinction in the definitions is that "distinct parts >= k" sets a lower bound to all parts, whereas "the least being ..." means that the lower limit must be attained by one of the parts. (End)

%C Comment from _N. J. A. Sloane_, Sep 28 2008: (Start) Generating functions and Maple programs for the sequences in the first and second columns of the above list are respectively:

%C For A025147, A025148, etc.:

%C f:=proc(k) product(1+x^j, j=k..100): series(%,x,100): seriestolist(%); end;

%C For A096765, A096749, etc.:

%C g:=proc(k) x^(k-1)*product(1+x^j, j=k..100): series(%,x,100): seriestolist(%); end; (End)

%C Also number of 2's in partitions of n+1 into distinct parts; also number of partitions of n+1 into distinct parts, the least being 1.

%C Number of different sums from 1+[1,3]+[1,4]+...+[1,n] - _Jon Perry_, Jan 01 2004

%C Also number of partitions of n such that if k is the largest part, then all parts from 1 to k occur, k occurring at least twice. Example: a(7)=3 because we have [2,2,2,1],[2,2,1,1,1] and [1,1,1,1,1,1,1]. - _Emeric Deutsch_, Apr 09 2006

%C Also number of partitions of n+1 such that if k is the largest part, then all parts from 1 to k occur, k occurring exactly once. Example: a(7)=3 because we have [3,2,2,1],[3,2,1,1,1] and [2,1,1,1,1,1,1] (there is a simple bijection with the partitions defined before). - _Emeric Deutsch_, Apr 09 2006

%C Also number of partitions of n+1 into distinct parts where the number of parts is itself a part. - _Reinhard Zumkeller_, Nov 04 2007

%C Partial sums give A038348 (observed by Jonathan Vos Post, proved by several correspondents).

%D Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997. MathEduc Database (Zentralblatt MATH, 1997c.01891).

%D Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.

%H INRIA Algorithms Project, <a href="http://algo.inria.fr/ecs/ecs?searchType=1&amp;service=Search&amp;searchTerms=798">Encyclopedia of Combinatorial Structures 798</a>

%H Reinhard Zumkeller and Alois P. Heinz, <a href="/A025147/b025147.txt">Table of n, a(n) for n = 0..1000</a> (terms n = 0..100 from Reinhard Zumkeller)

%F G.f.: Product_{k=2..inf} (1+x^k).

%F a(n) = A000009(n)-a(n-1) =sum_{0<=k<=n}(-1)^k*A000009(n-k) - _Henry Bottomley_, May 09 2002

%F a(n)=t(n, 1), where t(n, k)=1+Sum(t(i, j): i>j>k & i+j=n), 2<=k<=n. - _Reinhard Zumkeller_, Jan 01 2003

%F G.f.=1+sum(x^(k(k+3)/2)/product(1-x^j, j=1..k), k=1..infinity). - _Emeric Deutsch_, Apr 09 2006

%F The previous g.f. is a special case of the g.f. for partitions into distinct parts >= L, sum(n>=0, x^(n*(n+2*L-1)/2) / prod(k=1..n, 1-x^k) ). [_Joerg Arndt_, Mar 24 2011]

%F G.f.: sum(n>=1, x^(n*(n+1)/2-1) / prod(k=1..n-1, 1-x^k) ), a special case of the g.f. for partitions into distinct parts >= L, sum(n>=L-1, x^(n*(n+1)/2-L*(L-1)/2) / prod(k=1..n-(L-1), 1-x^k) ). [_Joerg Arndt_, Mar 27 2011]

%F a(n) = Sum(A060016(n-k+1,k-1): 1<k<=floor((n+2)/2) for n>0. - _Reinhard Zumkeller_, Nov 04 2007

%F a(n)=A096765(n+1). - _R. J. Mathar_, Jul 31 2008

%e a(7) = 3, from {{3, 4}, {2, 5}, {7}}

%p g:=product(1+x^j,j=2..65): gser:=series(g,x=0,62): seq(coeff(gser,x,n),n=0..57); - _Emeric Deutsch_, Apr 09 2006

%p with(combstruct):ZL := {L = PowerSet(Sequence(Z,card>=2)) },unlabeled:seq(count([L,ZL],size=i),i=0..57); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 09 2007

%t CoefficientList[Series[Product[1+q^n, {n, 2, 60}], {q, 0, 60}], q]

%t FoldList[ PartitionsQ[ #2+1 ]-#1&, 0, Range[ 64 ] ]

%o a025147 = p 2 where

%o p _ 0 = 1

%o p k m = if m < k then 0 else p (k + 1) (m - k) + p (k + 1) m

%o -- _Reinhard Zumkeller_, Dec 28 2011

%o (PARI) a(n)=if(n,my(v=partitions(n));sum(i=1,#v,v[i][1]>1&&v[i]==vecsort(v[i],,8)),1) \\ _Charles R Greathouse IV_, Nov 20 2012

%Y Cf. A015744, A015745, A015746, A015750, A015753, A015754, A015755, A002865.

%K nonn,easy,nice

%O 0,6

%A _Clark Kimberling_

%E Corrected and extended by Dean Hickerson (dean.hickerson(AT)yahoo.com), Oct 10, 2001

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