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A002865 Number of partitions of n that do not contain 1 as a part.
(Formerly M0309 N0113)
132
1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 41, 55, 66, 88, 105, 137, 165, 210, 253, 320, 383, 478, 574, 708, 847, 1039, 1238, 1507, 1794, 2167, 2573, 3094, 3660, 4378, 5170, 6153, 7245, 8591, 10087, 11914, 13959, 16424, 19196, 22519, 26252, 30701 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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, Feb 18 2009]

a(n) = A116449(n) + A116450(n). - Reinhard Zumkeller, 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. - Franklin T. Adams-Watters, May 01 2006

Number of partitions where the largest part occurs at least twice. - Joerg Arndt, Apr 17 2011

Row sums of triangle A147768. [Gary W. Adamson, Nov 11 2008]

From Lewis Mammel, 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)

Convolution product (1, 1, 2, 2, 4, 4,...) * (1, 2, 3,...) = A058682 starting (1, 3, 7, 13, 23, 37,...); with row sums of triangle A171239 = A058682. - Gary W. Adamson, Dec 05 2009

Also the number of 2-regular multigraphs with loops forbidden.

Number of appearances of the multiplicity n, n-1, ... , n-k in all partitions of n, for k < n/2.  (Only populated by multiplicities of large numbers of 1s.) [William Keith, Nov 20 2011]

a(n) = A090824(n,1) for n > 0. - Reinhard Zumkeller, Oct 10 2012

Also the number of equivalence classes of n X n binary matrices with exactly 2 1's in each row and column, up to permutations of rows and columns (cf. A133687). N. J. A. Sloane, Sep 16 2013

The q-Catalan numbers ((1-q)/(1-q^(n+1)))[2n,n]_q, where [2n,n]_q are the central q-binomial coefficients, match this sequence in their initial segment of length n. - William J. Keith, Nov 14 2013

REFERENCES

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).

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

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

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

LINKS

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].

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 100

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

R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv math.CO.0606404.

Index entries for related partition-counting sequences

FORMULA

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 A. Sellers, Dec 06 2005.

a(n) = Sum(A008284(n-k+1,k-1): 1<k<=floor((n+2)/2) for n>0. - Reinhard Zumkeller, Nov 04 2007

G.f.: 1 + sum(n>=2, x^n / prod(k>=n, 1-x^k)).  - Joerg Arndt, Apr 13 2011

G.f.: sum(n>=0, x^(2*n) / prod(k=1..n, 1-x^k ) ). - Joerg Arndt, Apr 17 2011

EXAMPLE

a(6) = 4 from 6 = 4+2 = 3+3 = 2+2+2.

G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 7*x^8 + 8*x^9 + ...

MAPLE

with(combstruct): ZL1:=[S, {S=Set(Cycle(Z, card>1))}, unlabeled]: seq(count(ZL1, size=n), n=0..50);  # Zerinvary Lajos, 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, 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, Jun 11 2008

MATHEMATICA

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

f[1, 1] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == n, 1, f[n, k + 1] + f[n - k, k]]]]; Table[ f[n, 2], {n, 50}] (* Robert G. Wilson v *)

PROG

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

(PARI) a(n)=if(n, numbpart(n)-numbpart(n-1), 1) \\ Charles R Greathouse IV, Nov 26 2012

(MAGMA) A41 := func<n|n ge 0 select NumberOfPartitions(n) else 0>; [A41(n)-A41(n-1):n in [0..50]];

CROSSREFS

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, A058682, A171239.

A column of A090824 and of A133687. Cf. A229161.

2-regular not necessarily connected graphs: A008483 (simple graphs), A000041 (multigraphs with loops allowed), this sequence (multigraphs with loops forbidden), A027336 (graphs with loops allowed but no multiple edges).

Sequence in context: A240019 A036000 * A085811 A187219 A014810 A239835

Adjacent sequences:  A002862 A002863 A002864 * A002866 A002867 A002868

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Regular graphs comment and cross references, and MAGMA code from Jason Kimberley, Jan 05 2011

STATUS

approved

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Last modified October 25 03:21 EDT 2014. Contains 248517 sequences.