Search: id:A025147
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%I A025147
%S A025147 1,0,1,1,1,2,2,3,3,5,5,7,8,10,12,15,17,21,25,29,35,41,48,56,66,76,89,103,
%T A025147 119,137,159,181,209,239,273,312,356,404,460,522,591,669,757,853,963,1085,
%U A025147 1219,1371,1539,1725,1933,2164,2418,2702,3016,3362,3746,4171,4637,5155
%N A025147 Number of partitions of n into distinct parts >= 2.
%C A025147 Comment from R. J. Mathar, Jul 31 2008: (Start) 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 A025147, A096765 (k=2)
%C A025147 A025148, A096749 (k=3)
%C A025147 A025149, A026824 (k=4)
%C A025147 A025150, A026825 (k=5)
%C A025147 A025151, A026826 (k=6)
%C A025147 A025152, A026827 (k=7)
%C A025147 A025153, A026828 (k=8)
%C A025147 A025154, A026829 (k=9)
%C A025147 A025155, A026830 (k=10)
%C A025147 A096740, A026831 (k=11)
%C A025147 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 A025147 Comment from N. J. A. Sloane (njas(AT)research.att.com), 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 A025147 For A025147, A025148, etc.:
%C A025147 f:=proc(k) product(1+x^j, j=k..100): series(%,x,100): seriestolist(%); 
               end;
%C A025147 For A096765, A096749, etc.:
%C A025147 g:=proc(k) x^(k-1)*product(1+x^j, j=k..100): series(%,x,100): seriestolist(%); 
               end; (End)
%C A025147 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 A025147 Number of different sums from 1+[1,3]+[1,4]+...+[1,n] - Jon Perry (perry(AT)globalnet.co.uk), 
               Jan 01 2004
%C A025147 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 (deutsch(AT)duke.poly.edu), Apr 09 2006
%C A025147 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 (deutsch(AT)duke.poly.edu), Apr 09 2006
%C A025147 Also number of partitions of n+1 into distinct parts where the number 
               of parts is itself a part. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Nov 04 2007
%H A025147 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=798">
               Encyclopedia of Combinatorial Structures 798</a>
%F A025147 G.f.: Product_{k=2..inf} (1+x^k).
%F A025147 a(n) = A000009(n)-a(n-1) =sum_{0<=k<=n}(-1)^k*A000009(n-k) - Henry Bottomley 
               (se16(AT)btinternet.com), May 09 2002
%F A025147 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 (reinhard.zumkeller(AT)gmail.com), Jan 01 2003
%F A025147 G.f.=1+sum(x^(k(k+3)/2)/product(1-x^j, j=1..k), k=1..infinity). - Emeric 
               Deutsch (deutsch(AT)duke.poly.edu), Apr 09 2006
%F A025147 a(n) = Sum(A060016(n-k+1,k-1): 1<k<=floor((n+2)/2) for n>0. - Reinhard 
               Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 04 2007
%F A025147 a(n)=A096765(n+1). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 
               31 2008
%F A025147 G.f.: product_{j=2..infinity} (1+x^j). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Jul 31 2008
%e A025147 a(7) = 3, from {{3, 4}, {2, 5}, {7}}
%p A025147 g:=product(1+x^j,j=2..65): gser:=series(g,x=0,62): seq(coeff(gser,x,n),
               n=0..57); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 09 2006
%p A025147 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 A025147 CoefficientList[Series[Product[1+q^n, {n, 2, 60}], {q, 0, 60}], q]
%t A025147 FoldList[ PartitionsQ[ #2+1 ]-#1&, 0, Range[ 64 ] ]
%Y A025147 Cf. A015744 A015745 A015746 A015750 A015753 A015754 A015755.
%Y A025147 Cf. A002865.
%Y A025147 Sequence in context: A119620 A029018 A096765 this_sequence A032230 A126793 
               A069910
%Y A025147 Adjacent sequences: A025144 A025145 A025146 this_sequence A025148 A025149 
               A025150
%K A025147 nonn,easy,nice
%O A025147 0,6
%A A025147 Clark Kimberling (ck6(AT)evansville.edu)
%E A025147 Corrected and extended by Dean Hickerson (dean.hickerson(AT)yahoo.com), 
               Oct 10, 2001

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