0,6

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:

A025147, A096765 (k=2)

A025148, A096749 (k=3)

A025149, A026824 (k=4)

A025150, A026825 (k=5)

A025151, A026826 (k=6)

A025152, A026827 (k=7)

A025153, A026828 (k=8)

A025154, A026829 (k=9)

A025155, A026830 (k=10)

A096740, A026831 (k=11)

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)

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:

For A025147, A025148, etc.:

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

For A096765, A096749, etc.:

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

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.

Number of different sums from 1+[1,3]+[1,4]+...+[1,n] - Jon Perry (perry(AT)globalnet.co.uk), Jan 01 2004

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

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

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 798

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

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

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

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

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

a(n)=A096765(n+1). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008

G.f.: product_{j=2..infinity} (1+x^j). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2008

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

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

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

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

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

Cf. A015744 A015745 A015746 A015750 A015753 A015754 A015755.

Cf. A002865.

Sequence in context: A119620 A029018 A096765 this_sequence A032230 A126793 A069910

Adjacent sequences: A025144 A025145 A025146 this_sequence A025148 A025149 A025150

nonn,easy,nice

Clark Kimberling (ck6(AT)evansville.edu)

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