0,5

Also the number of partitions of n-1, n>=2, such that the least part occurs exactly once. See A096373, A097091, A097092, A097093. - Robert G. Wilson v Jul 24 2004 [Corrected by Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Feb 18 2009]

a(n) = A116449(n) + A116450(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 16 2006

Number of partitions of n+1 where the number of parts is itself a part. Take a partition of n (with k parts) which does not contain 1, remove 1 from each part and add a new part of size k+1. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), May 01 2006

Equals row sums of triangle A147768 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 11 2008]

Contribution from Lewis Mammel (l_mammel(AT)att.net), Oct 06 2009: (Start)

a(n) is the number of sets of n disjoint pairs of 2n things,

called a pairing, disjoint with a given pairing ( A053871, )

that are unique under permutations preserving the given pairing.

Can be seen immediately from a graphical representation which must

decompose into even numbered cycles of 4 or more things, as connected

by pairs alternating between the pairings. Each thing is in a single cycle,

so this is a partition of 2n into even parts greater than 2,

equivalent to a partition of n into parts greater than 1. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 836.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, p*(n).

H. Gropp, On tactical configurations, regular bipartite graphs and (v,k,even)-designs, Discr. Math., 155 (1996), 81-98.

P. G. Tait, Scientific Papers, Cambridge Univ. Press, Vol. 1, 1898, Vol. 2, 1900, see Vol. 1, p. 334.

T. D. Noe, Table of n, a(n) for n=0..1000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 100

J. L. Nicolas and A. Sarkozy, On partitions without small parts

Index entries for related partition-counting sequences

G.f.: Product_{m>1} 1/(1-x^m).

a(0)=1, a(n)= p(n)-p(n-1), n>=1, with the partition numbers p(n) := A000041(n).

a(n) = A085811(n+2). - James Sellers, Dec 06 2005.

a(n) = Sum(A008284(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(6) = 4 from 6 = 4+2 = 3+3 = 2+2+2.

spec := [ B, {B=Set(Set(Z, card>1))}, unlabeled ]; [seq(combstruct[count](spec, size=n), n=1..50)];

with(combstruct):ZL1:=[S, {S=Set(Cycle(Z, card>1))}, unlabeled]:seq(count(ZL1, size=n), n=0..50); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 24 2007

G:={P=Set(Set(Atom, card>1))}:combstruct[gfsolve](G, unlabeled, x):seq(combstruct[count]([P, G, unlabeled], size=i), i=0..50); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 16 2007

with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, unlabeled]; end: A:=a(2):seq(count(A, size=n), n=0..50); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 11 2008

Table[ PartitionsP[n + 1] - PartitionsP[n], {n, -1, 50}] (from Robert G. Wilson v Jul 24 2004)

(PARI) a(n)=if(n<0, 0, polcoeff((1-x)/eta(x+x*O(x^n)), n))

First differences of partition numbers A000041. Cf. A053445, A072380, A081094, A081095.

Pairwise sums seem to be in A027336.

Essentially the same as A085811.

Cf. A025147.

A147768 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 11 2008]

Sequence in context: A035979 A035989 A036000 this_sequence A085811 A014810 A026929

Adjacent sequences: A002862 A002863 A002864 this_sequence A002866 A002867 A002868

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).