A008483 -id:A008483 - cp's OEIS frontend

A002865 Number of partitions of n that do not contain 1 as a part.

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

Offset: 0

N. J. A. Sloane

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]
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 (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)
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. - Jason Kimberley, Jan 05 2011
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 1's.) - William Keith, Nov 20 2011
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
Starting at a(2) this sequence gives the number of vertices on a nim tree created in the game of edge removal for a path P_{n} where n is the number of vertices on the path. This is the number of nonisomorphic graphs that can result from the path when the game of edge removal is played. - Lyndsey Wong, Jul 09 2016
The number of different ways to climb a staircase taking at least two stairs at a time. - Mohammad K. Azarian, Nov 20 2016
Let 1,0,1,1,1,... (offset 0) count unlabeled, connected, loopless 1-regular digraphs. This here is the Euler transform of that sequence, counting unlabeled loopless 1-regular digraphs. A145574 is the associated multiset transformation. A000166 are the labeled loopless 1-regular digraphs. - R. J. Mathar, Mar 25 2019
For n > 1, also the number of partitions with no part greater than the number of ones. - George Beck, May 09 2019 [See A187219 which is the correct sequence for this interpretation for n >= 1. - Spencer Miller, Jan 30 2023]
From Gus Wiseman, May 19 2019: (Start)
Conjecture: Also the number of integer partitions of n - 1 that have a consecutive subsequence summing to each positive integer from 1 to n - 1. For example, (32211) is such a partition because we have consecutive subsequences:
  1: (1)
  2: (2)
  3: (3) or (21)
  4: (22) or (211)
  5: (32) or (221)
  6: (2211)
  7: (322)
  8: (3221)
  9: (32211)
(End)
There is a sufficient and necessary condition to characterize the partitions defined by Gus Wiseman. It is that the largest part must be less than or equal to the number of ones plus one. Hence, the number of partitions of n with no part greater than the number of ones is the same as the number of partitions of n-1 that have a consecutive subsequence summing to each integer from 1 to n-1. Gus Wiseman's conjecture can be proved bijectively. - Andrew Yezhou Wang, Dec 14 2019
From Peter Bala, Dec 01 2024: (Start)
Let P(2, n) denote the set of partitions of n into parts k > 1. Then A000041(n) = - Sum_{parts k in all partitions in P(2, n+2)} mu(k). For example, with n = 5, there are 4 partitions of n + 2 = 7 into parts greater than 1, namely, 7, 5 + 2, 4 + 3, 3 + 2 + 2, and mu(7) + (mu(5) + mu(2)) + (mu(4 ) + mu(3)) + (mu(3) + mu(2) + mu(2)) = -7 = - A000041(5). (End)

			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 + ...
From _Gus Wiseman_, May 19 2019: (Start)
The a(2) = 1 through a(9) = 8 partitions not containing 1 are the following. The Heinz numbers of these partitions are given by A005408.
  (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)
            (22)  (32)  (33)   (43)   (44)    (54)
                        (42)   (52)   (53)    (63)
                        (222)  (322)  (62)    (72)
                                      (332)   (333)
                                      (422)   (432)
                                      (2222)  (522)
                                              (3222)
The a(2) = 1 through a(9) = 8 partitions of n - 1 whose least part appears exactly once are the following. The Heinz numbers of these partitions are given by A247180.
  (1)  (2)  (3)   (4)   (5)    (6)    (7)     (8)
            (21)  (31)  (32)   (42)   (43)    (53)
                        (41)   (51)   (52)    (62)
                        (221)  (321)  (61)    (71)
                                      (331)   (332)
                                      (421)   (431)
                                      (2221)  (521)
                                              (3221)
The a(2) = 1 through a(9) = 8 partitions of n + 1 where the number of parts is itself a part are the following. The Heinz numbers of these partitions are given by A325761.
  (21)  (22)  (32)   (42)   (52)    (62)    (72)     (82)
              (311)  (321)  (322)   (332)   (333)    (433)
                            (331)   (431)   (432)    (532)
                            (4111)  (4211)  (531)    (631)
                                            (4221)   (4222)
                                            (4311)   (4321)
                                            (51111)  (4411)
                                                     (52111)
The a(2) = 1 through a(8) = 7 partitions of n whose greatest part appears at least twice are the following. The Heinz numbers of these partitions are given by A070003.
  (11)  (111)  (22)    (221)    (33)      (331)      (44)
               (1111)  (11111)  (222)     (2221)     (332)
                                (2211)    (22111)    (2222)
                                (111111)  (1111111)  (3311)
                                                     (22211)
                                                     (221111)
                                                     (11111111)
Nonisomorphic representatives of the a(2) = 1 through a(6) = 4 2-regular multigraphs with n edges and n vertices are the following.
  {12,12}  {12,13,23}  {12,12,34,34}  {12,12,34,35,45}  {12,12,34,34,56,56}
                       {12,13,24,34}  {12,13,24,35,45}  {12,12,34,35,46,56}
                                                        {12,13,23,45,46,56}
                                                        {12,13,24,35,46,56}
The a(2) = 1 through a(9) = 8 partitions of n with no part greater than the number of ones are the following. The Heinz numbers of these partitions are given by A325762.
  (11)  (111)  (211)   (2111)   (2211)    (22111)    (22211)     (33111)
               (1111)  (11111)  (3111)    (31111)    (32111)     (222111)
                                (21111)   (211111)   (41111)     (321111)
                                (111111)  (1111111)  (221111)    (411111)
                                                     (311111)    (2211111)
                                                     (2111111)   (3111111)
                                                     (11111111)  (21111111)
                                                                 (111111111)
(End)

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. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
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.

Andrew van den Hoeven, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
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].
A. P. Akande et al., Computational study of non-unitary partitions, arXiv:2112.03264 [math.CO], 2021.
Colin Albert, Olivia Beckwith, Irfan Demetoglu, Robert Dicks, John H. Smith, and Jasmine Wang, Integer partitions with large Dyson rank, arXiv:2203.08987 [math.NT], 2022.
Max A. Alekseyev and Allan Bickle, Forbidden Subgraphs of Single Graphs, (2024). See p. 6.
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
G. Dahl and T. A. Haufmann, Zero-one completely positive matrices and the A(R,S) classes, Preprint, 2016.
Atul Dixit, Gaurav Kumar, and Aviral Srivastava, Non-Rascoe partitions and a rank parity function associated to the Rogers-Ramanujan partitions, arXiv:2508.04359 [math.CO], 2025. See references.
R. P. Gallant, G. Gunther, B. L. Hartnell, and D. F. Rall, A game of edge removal on graphs, JCMCC, 57 (2006), 75 - 82.
Edray Herber Goins and Talitha M. Washington, On the generalized climbing stairs problem, Ars Combin. 117 (2014), 183-190. MR3243840 (Reviewed), arXiv:0909.5459 [math.CO], 2009.
H. Gropp, On tactical configurations, regular bipartite graphs and (v,k,even)-designs, Discr. Math., 155 (1996), 81-98.
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Cristiano Husu, The butterfly sequence: the second difference sequence of the numbers of integer partitions with distinct parts, its pentagonal number structure, its combinatorial identities and the cyclotomic polynomials 1-x and 1+x+x^2, arXiv:1804.09883 [math.NT], 2018.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 100
Wenwei Li, Approximation of the Partition Number After Hardy and Ramanujan: An Application of Data Fitting Method in Combinatorics, arXiv preprint arXiv:1612.05526 [math.NT], 2016-2018.
Wenwei Li, On the Number of Conjugate Classes of Derangements, arXiv:1612.08186 [math.CO], 2016.
J. L. Nicolas and A. Sárközy, On partitions without small parts, Journal de théorie des nombres de Bordeaux, 12 no. 1 (2000), p. 227-254.
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
H. P. Robinson, Letter to N. J. A. Sloane, Jul 12 1971
H. P. Robinson, Letter to N. J. A. Sloane, Dec 10 1973
H. P. Robinson, Letter to N. J. A. Sloane, Jan 4 1974.
Noah Rubin, Curtis Bright, Kevin K. H. Cheung, and Brett Stevens, Integer and Constraint Programming Revisited for Mutually Orthogonal Latin Squares, arXiv:2103.11018 [cs.DM], 2021.
Miloslav Znojil, Non-Hermitian N-state degeneracies: unitary realizations via antisymmetric anharmonicities, arXiv:2010.15014 [quant-ph], 2020.
Miloslav Znojil, Quantum phase transitions mediated by clustered non-Hermitian degeneracies, arXiv:2102.12272 [quant-ph], 2021.
Miloslav Znojil, Bose-Einstein condensation processes with nontrivial geometric multiplicites realized via PT-symmetric and exactly solvable linear-Bose-Hubbard building blocks, arXiv:2108.07110 [quant-ph], 2021.
Index entries for related partition-counting sequences

First differences of partition numbers A000041. Cf. A053445, A072380, A081094, A081095, A232697.
Pairwise sums seem to be in A027336.
Essentially the same as A085811.
Cf. A025147, A147768, A058682, A171239.
A column of A090824 and of A133687 and of A292508 and of A292622. 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). - Jason Kimberley, Jan 05 2011
See also A098743 (parts that do not divide n).
Numbers n such that in the edge-delete game on the path P_{n} the first player does not have a winning strategy: A274161. - Lyndsey Wong, Jul 09 2016
Cf. A002033, A070003, A103295, A247180, A325676, A325684, A325761, A325762, A325763.
Row sums of characteristic array A145573.
Number of partitions of n into parts >= m: A008483 (m = 3), A008484 (m = 4), A185325 - A185329 (m = 5 through 9).

Concatenation([1],List([1..41],n->NrPartitions(n)-NrPartitions(n-1))); # Muniru A Asiru, Aug 20 2018

A41 := func; [A41(n)-A41(n-1):n in [0..50]]; // Jason Kimberley, Jan 05 2011

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
# alternative Maple program:
A002865:= proc(n) option remember; `if`(n=0, 1, add(
      (numtheory[sigma](j)-1)*A002865(n-j), j=1..n)/n)
    end:
seq(A002865(n), n=0..60);  # Alois P. Heinz, Sep 17 2017

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 *)
Table[SeriesCoefficient[Exp[Sum[x^(2*k)/(k*(1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 50}] (* Vaclav Kotesovec, Aug 18 2018 *)
CoefficientList[Series[1/QPochhammer[x^2, x], {x,0,50}], x] (* G. C. Greubel, Nov 03 2019 *)
Table[Count[IntegerPartitions[n],?(FreeQ[#,1]&)],{n,0,50}] (* _Harvey P. Dale, Feb 12 2023 *)

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

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

from sympy import npartitions
def A002865(n): return npartitions(n)-npartitions(n-1) if n else 1 # Chai Wah Wu, Mar 30 2023

def A002865_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product((1-x^(m+2)) for m in (0..60)) ).list()
A002865_list(50) # G. C. Greubel, Nov 03 2019

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+3). - James Sellers, Dec 06 2005 [Corrected by Gionata Neri, Jun 14 2015]
a(n) = A116449(n) + A116450(n). - Reinhard Zumkeller, Feb 16 2006
a(n) = Sum_{k=2..floor((n+2)/2)} A008284(n-k+1,k-1) for n > 0. - Reinhard Zumkeller, Nov 04 2007
G.f.: 1 + Sum_{n>=2} x^n / Product_{k>=n} (1 - x^k). - Joerg Arndt, Apr 13 2011
G.f.: Sum_{n>=0} x^(2*n) / Product_{k=1..n} (1 - x^k). - Joerg Arndt, Apr 17 2011
a(n) = A090824(n,1) for n > 0. - Reinhard Zumkeller, Oct 10 2012
a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (12*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + 13*Pi/(24*sqrt(6)))/sqrt(n) + (217*Pi^2/6912 + 9/(2*Pi^2) + 13/8)/n). - Vaclav Kotesovec, Feb 26 2015, extended Nov 04 2016
G.f.: exp(Sum_{k>=1} (sigma_1(k) - 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018
a(0) = 1, a(n) = A232697(n) - 1. - George Beck, May 09 2019
From Peter Bala, Feb 19 2021: (Start)
G.f.: A(q) = Sum_{n >= 0} q^(n^2)/( (1 - q)*Product_{k = 2..n} (1 - q^k)^2 ).
More generally, A(q) = Sum_{n >= 0} q^(n*(n+r))/( (1 - q) * Product_{k = 2..n} (1 - q^k)^2 * Product_{i = 1..r} (1 - q^(n+i)) ) for r = 0,1,2,.... (End)
G.f.: 1 + Sum_{n >= 1} x^(n+1)/Product_{k = 1..n-1} 1 - x^(k+2). - Peter Bala, Dec 01 2024

A026794 Triangular array T read by rows: T(n,k) = number of partitions of n in which least part is k, 1<=k<=n.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 5, 1, 0, 0, 1, 7, 2, 1, 0, 0, 1, 11, 2, 1, 0, 0, 0, 1, 15, 4, 1, 1, 0, 0, 0, 1, 22, 4, 2, 1, 0, 0, 0, 0, 1, 30, 7, 2, 1, 1, 0, 0, 0, 0, 1, 42, 8, 3, 1, 1, 0, 0, 0, 0, 0, 1, 56, 12, 4, 2, 1, 1, 0, 0, 0, 0, 0, 1, 77, 14, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1, 101, 21, 6, 3, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Clark Kimberling

At least one part is k and each part is at least k.
From Emeric Deutsch, Feb 19 2006: (Start)
Also number of partitions of n in which the largest part occurs exactly k times. Example: T(6,2)=2 because we have [3,3] and [2,2,1,1].
G.f. of column k is x^k/prod(j>=k, 1-x^j ) (k>=1).
Row sums yield the partition numbers (A000041).
T(n,1) = A000041(n-1) (the partition numbers).
T(n,2) = A002865(n-2) (n>=2).
T(n,3)=A026796(n). T(n,4) = A026797(n). T(n,5) = A026798(n). T(n,6) = A026799(n). T(n,7) = A026800(n). T(n,8) = A026801(n). T(n,9) = A026802(n). T(n,10) = A026803(n).
Sum(k*T(n,k),k=1..n) = A046746(n). (End)
Triangle inverse = A161363. - Gary W. Adamson, Jun 07 2009
T(n,g) is also the number of not necessarily connected 2-regular graphs with girth exactly g: the part i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles. - Jason Kimberley, Feb 05 2012
From Bob Selcoe, Jul 24 2014 (Start):
Below is a process to generate equations for column k.
Let P be the partition numbers A000041(n-j) and let f(k) denote equations which generate column k.
To find f(k), start with f(1) = P(n-j), j=1. Thus T(n,1) = f(1) = P(n-1). This is the equation for column 1.
To find f(k) k>1, first sum the terms of f(k-1) replacing the value j with j+1, and then subtract the terms of f(k-1) replacing the value j with j+k. So to find f(2) (i.e., the equation for column 2, where k=2), start with f(1) = P(n-1); first replace j with j+1 (yielding P(n-2)), and then replace j with j+2 (yielding P(n-3)).  Subtracting the second term from the first, we get: f(2) = P(n-2) - P(n-3).
To find f(3), start with f(2), replace j with j+1 (yielding P(n-3) - P(n-4)) and then replace j with j+3 (yielding P(n-5) - P(n-6)). Subtracting the second group of terms from the first, we get: f(3) = P(n-3) - P(n-4) - P(n-5) + P(n-6). This is the equation for column 3; also the equation for T(n,3) = A026796(n). So for example, T(13,3) = 5 because P(13-3) - P(13-4) - P(13-5) + P(13-6) = 42 - 30 - 22 + 15 = 5.
Continue as above to find f(k) k={4..inf.}.  This will generate equations for T(n,4) = A026797(n), T(n,5) = A026798(n), T(n,6) = A026799(n), ad inf.
(End)

			T(12,3) = 4 because we have [9,3], [6,3,3], [5,4,3] and [3,3,3,3]. - Edited by _Bob Selcoe_, Sep 03 2016
Triangle starts:
    1;
    1,  1;
    2,  0, 1;
    3,  1, 0, 1;
    5,  1, 0, 0, 1;
    7,  2, 1, 0, 0, 1;
   11,  2, 1, 0, 0, 0, 1;
   15,  4, 1, 1, 0, 0, 0, 1;
   22,  4, 2, 1, 0, 0, 0, 0, 1;
   30,  7, 2, 1, 1, 0, 0, 0, 0, 1;
   42,  8, 3, 1, 1, 0, 0, 0, 0, 0, 1;
   56, 12, 4, 2, 1, 1, 0, 0, 0, 0, 0, 1;
   77, 14, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1;
  101, 21, 6, 3, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1;
  135, 24, 9, 3, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Kevin Brown, On Euler's Pentagonal Theorem, 1994-2008.
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g.
Johannes W. Meijer, Euler's ship on the Pentagonal Sea, pdf and jpg.
J. W. Meijer and M. Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, Volume 4, No. 1, December 2008. pp. 176-187.
Tilman Piesk, First 50 rows and illustrations of columns n = 2, 3, 4, 5, 6, 7, 8. a(n, k) is the number of gray fields in row n of table k.

Row sums give A000041.
Cf. A000041, A046746, A008284, A161363, A161364, A238341.
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9). For g >= 3, girth at least g implies no loops or parallel edges. - Jason Kimberley, Feb 05 2012
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: this sequence (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Feb 05 2012

g:=sum(t^i*x^i/product(1-x^j,j=i..30),i=1..30): gser:=simplify(series(g,x=0,19)): for n from 1 to 15 do P[n]:=coeff(gser,x^n) od: for n from 1 to 15 do seq(coeff(P[n],t^j),j=1..n) od; # Emeric Deutsch, Feb 19 2006
nmax:=13; for n from 1 to nmax do T(n,n):=1 od: for n from 1 to nmax do for k from floor(n/2)+1 to n-1 do T(n,k):=0 od: od: for n from 2 to nmax do for k from 1 to floor(n/2) do T(n,k):=sum(T(n-k,i),i=k..n-k) od: od: seq(seq(T(n,k),k=1..n), n=1..nmax); # Johannes W. Meijer, Jun 21 2010
nmax:=13; with(combinat): for n from 1 to nmax do for k from n+1 to nmax do T(n,k):=0 od: od: for n from 1 to nmax do T(n,1):=numbpart(n-1) od: for n from 1 to nmax do T(n,n):=1 od: for n from 2 to nmax do for k from 2 to n-1 do T(n,k) := T(n-1,k-1) - T(n-k,k-1) od: od: seq(seq(T(n,k),k=1..n), n=1..nmax); # Johannes W. Meijer, Jun 21 2010
#
p:= (f, g)-> zip((x,y)-> x+y, f, g, 0):
b:= proc(n, i) option remember; local h;
      h:= `if`(n=i and i>0, [0$(i-1), 1], []);
      `if`(i<1, h, p(p(h, b(n, i-1)), `if`(n b(n, n)[]:
seq(T(n), n=1..14); # Alois P. Heinz, Mar 28 2012

t[n_, k_] /; k<1 || k>n = 0; t[n_, n_] = 1; t[n_, k_] := t[n, k] = Sum[t[n-k, i], {i, k, n-k}]; Flatten[ Table[t[n, k], {n, 1, 14}, {k, 1, n}]] (* Jean-François Alcover, May 11 2012, after PARI *)

{T(n, k) = if( k<1 || k>n, 0, if( n==k, 1, sum(i=k, n-k, T(n-k, i))))} \\ Michael Somos, Feb 06 2003

A026794(n,k)=#select(p->p[1]==k,partitions(n,[k,n])) \\ For illustration: Creates the list of all partitions of n with smallest part equal to k. - M. F. Hasler, Jun 14 2018

T(n, k) = sum{T(n-k, i), k<=i<=n-k} for k=1, 2, ..., m, T(n, k)=0 for k=m+1, ..., n-1, where m=floor(n/2); T(n, n)=1 for n >= 1.
G.f.: G(t,x)=sum(t^i*x^i/product(1-x^j, j=i..infinity), i=1..infinity). - Emeric Deutsch, Feb 19 2006
G.f.: Sum_{k>=1} tx^k/(1-tx^k)/product(1-x^j,j=1..k-1). - Emeric Deutsch, Mar 13 2006
T(n,k) = T(n-1,k-1) - T(n-k,k-1) for n>=2 and 2<=k<=(n-1) with T(n,1) = A000041(n-1), T(n,n) = 1 for n>=1 and T(n,k) = 0 for k>n. - Johannes W. Meijer, Jun 21 2010
T(k,k) = 1 and T(n,1) = row sum (n-1); thus Meijer's 2010 formula generates the triangle without a priori reference to A000041 (the partition sequence). - Bob Selcoe, Sep 03 2016

More terms from Emeric Deutsch, Feb 19 2006

A005176 Number of regular graphs with n unlabeled nodes.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 8, 6, 22, 26, 176, 546, 19002, 389454, 50314870, 2942198546, 1698517037030, 442786966117636, 649978211591622812, 429712868499646587714, 2886054228478618215888598, 8835589045148342277802657274, 152929279364927228928025482936226, 1207932509391069805495173417972533120, 99162609848561525198669168653641835566774

Offset: 0

N. J. A. Sloane

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

E. Friedman, Illustration of small graphs
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
Jennifer M. Larson, Cheating Because They Can: Social Networks and Norm Violators, 2014. See Footnote 11.
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Regular Graph.

Not necessarily connected simple regular graphs: A005176 (any degree), A051031 (triangular array), specified degree k: A000012 (k=0), A059841 (k=1), A008483 (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7), A180260 (k=8).
Simple regular graphs of any degree: A005177 (connected), A068932 (disconnected), this sequence (not necessarily connected).
Not necessarily connected regular simple graphs with girth at least g: this sequence (g=3), A185314 (g=4), A185315 (g=5), A185316 (g=6), A185317 (g=7), A185318 (g=8), A185319 (g=9).
Cf. A295193.

a(n) = A005177(n) + A068932(n). - David Wasserman, Mar 08 2002

Row sums of triangle A051031.

More terms from David Wasserman, Mar 08 2002
a(15) and a(16) from Jason Kimberley, Sep 25 2009
Edited by Jason Kimberley, Jan 06 2011 and May 24 2012
a(17)-a(21) from Andrew Howroyd, Mar 08 2020
a(22)-a(24) from Andrew Howroyd, Apr 05 2020

A027336 Number of partitions of n that do not contain 2 as a part.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 35, 45, 58, 75, 96, 121, 154, 193, 242, 302, 375, 463, 573, 703, 861, 1052, 1282, 1555, 1886, 2277, 2745, 3301, 3961, 4740, 5667, 6754, 8038, 9548, 11323, 13398, 15836, 18678, 22001, 25873, 30383, 35620, 41715, 48771

Offset: 0

Clark Kimberling

nonn

Pairwise sums of sequence A002865 (partitions in which the least part is at least 2).
Also number of partitions of n into parts with at most one 1. - Reinhard Zumkeller, Oct 25 2004
Also number of partitions of n into parts with at least half of the parts having size 1; equivalently (by duality) number of partitions of n where the large part is at least twice as big as the second largest part. - Franklin T. Adams-Watters, Jun 08 2005
Also number of 2-regular not necessarily connected graphs with loops allowed but no multiple edges. - Jason Kimberley, Jan 05 2011

Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.
V. Jelinek, T. Mansour, and M. Shattuck, On multiple pattern avoiding set partitions, Advances in Applied Mathematics Volume 50, Issue 2, February 2013, Pages 292-326. - _N. J. A. Sloane_, Jan 01 2013
Jerome Kelleher and Barry O'Sullivan, Generating All Partitions: A Comparison Of Two Encodings, arXiv:0909.2331 [cs.DS], 2009-2014. [_Peter Luschny_, Oct 24 2010]
Krishna Menon and Anurag Singh, Pattern avoidance and dominating compositions, arXiv:2104.07274 [math.CO], 2021.
Mircea Merca, Fast algorithm for generating ascending compositions, arXiv:1903.10797 [math.CO], 2019.

Cf. A000041, A002865, A027337.
2-regular not necessarily connected graphs: A008483 (simple graphs), A000041 (multigraphs with loops allowed), A002865 (multigraphs with loops forbidden), A027336 (graphs with loops allowed but no multiple edges). - Jason Kimberley, Jan 05 2011
Column k=1 of A292622.

A41 := func;
[A41(n)-A41(n-2):n in [0..49]]; // Jason Kimberley, Jan 05 2011

with(combinat): a:=proc(n) if n=0 then 1 elif n=1 then 1 else numbpart(n)-numbpart(n-2) fi end: seq(a(n),n=0..49); # Emeric Deutsch, Feb 18 2006

a[n_] = PartitionsP[n] - PartitionsP[n-2]; a /@ Range[0, 49] (* Jean-François Alcover, Jul 13 2011, after Emeric Deutsch *)

a(n)=if(n<0,0,polcoeff((1-x^2)/eta(x+x*O(x^n)),n))

G.f.: (1 - x^2)*Product_{m>=1} 1/(1 - x^m).
a(n) = A000041(n) - A000041(n-2).
a(n) = p(n) - p(n-2) for n >= 2, where p(n) are the partition numbers (A000041); follows at once from the g.f. - Emeric Deutsch, Feb 18 2006
a(n) ~ exp(sqrt(2*n/3)*Pi)*Pi / (6*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + 25*Pi/(24*sqrt(6)))/sqrt(n) + (25/8 + 9/(2*Pi^2) + 817*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 04 2016

More terms from Benoit Cloitre, Dec 10 2002

A005638 Number of unlabeled trivalent (or cubic) graphs with 2n nodes.

Original entry on oeis.org

1, 0, 1, 2, 6, 21, 94, 540, 4207, 42110, 516344, 7373924, 118573592, 2103205738, 40634185402, 847871397424, 18987149095005, 454032821688754, 11544329612485981, 310964453836198311, 8845303172513781271

Offset: 0

N. J. A. Sloane

Because the triangle A051031 is symmetric, a(n) is also the number of (2n-4)-regular graphs on 2n vertices.

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. Brinkmann, Fast generation of cubic graphs, Journal of Graph Theory, 23(2):139-149, 1996.
Jason Kimberley, Not-necessarily connected regular graphs
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
R. W. Robinson, Cubic graphs (notes)
Robinson, R. W.; Wormald, N. C., Numbers of cubic graphs, J. Graph Theory 7 (1983), no. 4, 463-467.
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Cubic Graph
Gal Weitz, Lirandë Pira, Chris Ferrie, and Joshua Combes, Sub-universal variational circuits for combinatorial optimization problems, arXiv:2308.14981 [quant-ph], 2023.

Cf. A000421.
Row sums of A275744.
3-regular simple graphs: A002851 (connected), A165653 (disconnected), this sequence (not necessarily connected).
Regular graphs A005176 (any degree), A051031 (triangular array), chosen degrees: A000012 (k=0), A059841 (k=1), A008483 (k=2), this sequence (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7), A180260 (k=8).
Not necessarily connected 3-regular simple graphs with girth *at least* g: this sequence (g=3), A185334 (g=4), A185335 (g=5), A185336 (g=6).
Not necessarily connected 3-regular simple graphs with girth *exactly* g: A185133 (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).

a(n) = A002851(n) + A165653(n).

This sequence is the Euler transformation of A002851.

More terms from Ronald C. Read.

Comment, formulas, and (most) crossrefs by Jason Kimberley, 2009 and 2012

A008484 Number of partitions of n into parts >= 4.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 11, 12, 16, 18, 24, 27, 34, 39, 50, 57, 70, 81, 100, 115, 140, 161, 195, 225, 269, 311, 371, 427, 505, 583, 688, 791, 928, 1067, 1248, 1434, 1668, 1914, 2223, 2546, 2945, 3370, 3889, 4443, 5113, 5834, 6698

Offset: 0

T. Forbes (anthony.d.forbes(AT)googlemail.com)

a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 4 (all such graphs are simple). The integer i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles. - Jason Kimberley, Jan 2011 and Feb 2012
By removing a single part of size 4, an A026797 partition of n becomes an A008484 partition of n - 4. Hence this sequence is essentially the same as A026797. - Jason Kimberley, Feb 2012
Number of partitions of n+3 such that 3*(number of parts) is a part. - Clark Kimberling, Feb 27 2014
Let c(n) be the number of partitions of n such that both (number of parts) and 2*(number of parts) are parts; then c(n) = a(n-6) for n >= 6 and c(n) = 0 for n < 6. - Clark Kimberling, Mar 01 2014
a(n) is also the number of partitions of n for which three times the number of ones is twice the number of parts (conjectured). - George Beck, Aug 19 2017
Proof: Above definition is equivalent to 2 out of 3 parts being equal to 1. Arrange in triples 1, 1, >= 2, etc. Sum of each triple corresponds to sequence definition. - Martin Fuller, Aug 21 2023

Giovanni Resta, Table of n, a(n) for n = 0..1000
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
Jason Kimberley, Not necessarily connected k-regular graphs with girth at least 4
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

2-regular graphs with girth at least 4: A185114 (connected), A185224 (disconnected), this sequence (not necessarily connected).
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), this sequence (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).
Not necessarily connected k-regular simple graphs with girth at least 4: A185314 (any k), A185304 (triangle); specified degree k: this sequence (k=2), A185334 (k=3), A185344 (k=4), A185354 (k=5), A185364 (k=6).

a:= func< n | NumberOfPartitions(n)-NumberOfPartitions(n-1)-NumberOfPartitions(n-2)+ NumberOfPartitions(n-4)+NumberOfPartitions(n-5)- NumberOfPartitions(n-6) >; [1,0,0,0,1,1,1] cat [ a(n) : n in [7..60]]; // Vincenzo Librandi, Aug 20 2017

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/(&*[1-x^(m+4): m in [0..70]]) )); // G. C. Greubel, Nov 03 2019

series(1/product((1-x^i),i=4..65),x,60); # end of program
ZL := [ B,{B=Set(Set(Z, card>=4))}, unlabeled ]: seq(combstruct[count](ZL, size=n), n=0..60); # Zerinvary Lajos, Mar 13 2007

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, 4], {n, 60}] (* end of program *)
Drop[Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 3*Length[p]]], {n, 60}],2]  (* Clark Kimberling, Feb 27 2014 *)
Table[Count[IntegerPartitions[n],
  p_ /; 3 Count[p, 1] == 2 Length[p]], {n, 0, 60}] (* George Beck Aug 19 2017 *)
CoefficientList[Series[1/QPochhammer[x^4, x], {x,0,60}], x] (* G. C. Greubel, Nov 03 2019 *)

my(x='x+O('x^60)); Vec(1/prod(m=0,70, 1-x^(m+4))) \\ G. C. Greubel, Nov 03 2019

def A008484_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product((1-x^(m+4)) for m in (0..70)) ).list()
A008484_list(60) # G. C. Greubel, Nov 03 2019

G.f.: 1 / Product_{m>=4} (1 - x^m).
Euler transformation of A185114. - Jason Kimberley, Jan 30 2011
Given by p(n) - p(n-1) - p(n-2) + p(n-4) + p(n-5) - p(n-6) where p(n) = A000041(n). Generally, 1/Product_{i>=K} (1 - x^i) is given by p({A}), where {A} is defined over the coefficients of Product_{i=1..K-1} (1 - x^i). In this case, K=4, so (1-x)(1-x^2)(1-x^3) = 1 - x - x^2 + x^4 + x^5 - x^6, defining {A} as above. G.f.: 1 + Sum_{i>=1} (x^4i)/Product_{j=1..i}(1 - x^j). - Jon Perry, Jul 04 2004
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^3 / (12*sqrt(2)*n^(5/2)). - Vaclav Kotesovec, Jun 02 2018
G.f.: exp(Sum_{k>=1} x^(4*k)/(k*(1 - x^k))). - Ilya Gutkovskiy, Aug 21 2018
G.f.: 1 + Sum_{n >= 1} x^(n+3)/Product_{k = 0..n-1} (1 - x^(k+4)). - Peter Bala, Dec 01 2024

A118096 Number of partitions of n such that the largest part is twice the smallest part.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 4, 6, 6, 6, 10, 9, 11, 13, 14, 15, 20, 18, 23, 25, 27, 27, 37, 35, 39, 43, 48, 49, 61, 57, 68, 72, 78, 81, 97, 95, 107, 114, 127, 128, 150, 148, 168, 179, 191, 198, 229, 230, 254, 266, 291, 300, 338, 344, 379, 398, 427, 444, 498, 505, 550, 580, 625

Offset: 1

Emeric Deutsch, Apr 12 2006

nonn

Also number of partitions of n such that if the largest part occurs k times, then the number of parts is 2k. Example: a(8)=4 because we have [7,1], [6,2], [5,3] and [3,3,1,1].

			a(8)=4 because we have [4,2,2], [2,2,2,1,1], [2,2,1,1,1,1] and [2,1,1,1,1,1,1].

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Cf. A237825, A237826, A237827.

Cf. A008483, A117086, A238479, A350893.

g:=sum(x^(3*k)/product(1-x^j,j=k..2*k),k=1..30): gser:=series(g,x=0,75): seq(coeff(gser,x,n),n=1..70);
# second Maple program:
b:= proc(n, i, t) option remember: `if`(n=0, 1, `if`(in, 0, b(n-i, i, t))))
    end:
a:= n-> add(b(n-3*j, 2*j, j), j=1..n/3):
seq(a(n), n=1..64);  # Alois P. Heinz, Sep 04 2017

Table[Count[IntegerPartitions[n], p_ /; 2 Min[p] = = Max[p]], {n, 40}] (* Clark Kimberling, Feb 16 2014 *)
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < t, 0,
     b[n, i - 1, t] + If[i > n, 0, b[n - i, i, t]]]];
a[n_] := Sum[b[n - 3j, 2j, j], {j, 1, n/3}];
Array[a, 64] (* Jean-François Alcover, Jun 04 2021, after Alois P. Heinz *)
(* Third program: *)
nmax = 100; p = 1; s = 0; Do[p = Simplify[p*(1 - x^(2*k - 1))*(1 - x^(2*k))/(1 - x^k)]; p = Normal[p + O[x]^(nmax+1)]; s += x^(3*k)/(1 - x^k)/p;, {k, 1, nmax}]; Rest[CoefficientList[Series[s, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 16 2025 *)

my(N=70, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/prod(j=k, 2*k, 1-x^j)))) \\ Seiichi Manyama, May 14 2023

G.f.: Sum_{k>=1} x^(3*k)/Product_{j=k..2*k} (1-x^j).

a(n) ~ exp(Pi*sqrt(2*n/15)) / (5^(1/4)*sqrt(2*phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 13 2025

A026796 Number of partitions of n in which the least part is 3.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 10, 13, 17, 21, 25, 33, 39, 49, 60, 73, 88, 110, 130, 158, 191, 230, 273, 331, 391, 468, 556, 660, 779, 927, 1087, 1284, 1510, 1775, 2075, 2438, 2842, 3323, 3872, 4510, 5237, 6095, 7056, 8182, 9465, 10945, 12625

Offset: 0

Clark Kimberling

Let b(k) be the number of partitions of k for which twice the number of ones is the number of parts, k = 0, 1, 2, ... . Then a(n+4) = b(n), n = 0, 1, 2, ... (conjectured). - George Beck, Aug 19 2017

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Essentially the same sequence as A008483.
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2 -- multigraphs with at least one pair of parallel edges, but loops forbidden), this sequence (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800 (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).
Not necessarily connected k-regular simple graphs girth exactly 3: A198313 (any k), A185643 (triangle); fixed k: this sequence (k=2), A185133 (k=3), A185143 (k=4), A185153 (k=5), A185163 (k=6).

R:=PowerSeriesRing(Integers(), 60); [0,0,0] cat Coefficients(R!( x^3/(&*[1-x^(m+3): m in [0..70]]) )); // G. C. Greubel, Nov 02 2019

seq(coeff(series(x^3/mul(1-x^(m+3), m=0..65), x, n+1), x, n), n = 0 .. 60); # G. C. Greubel, Nov 02 2019

Table[Count[IntegerPartitions[n], p_ /; Min@p==3], {n, 0, 60}] (* George Beck Aug 19 2017 *)
CoefficientList[Series[x^3/QPochhammer[x^3, x], {x,0,60}], x] (* G. C. Greubel, Nov 02 2019 *)

a(n) = numbpart(n-3) - numbpart(n-4) - numbpart(n-5) + numbpart(n-6); \\ Michel Marcus, Aug 20 2014

x='x+O('x^66); Vecrev(Pol(x^3*(1-x)*(1-x^2)/eta(x))) \\ Joerg Arndt, Aug 22 2014

def A026796_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^3/product((1-x^(m+3)) for m in (0..65)) ).list()
A026796_list(60) # G. C. Greubel, Nov 02 2019

G.f.: x^3 / Product_{m>=3} (1 - x^m).
a(n) = p(n-3) - p(n-4) - p(n-5) + p(n-6), where p(n) = A000041(n). - Bob Selcoe, Aug 07 2014
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^2 / (12*sqrt(3)*n^2). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=1} x^(3*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020

More terms from Michel Marcus, Aug 20 2014

a(0) = 0 prepended by Joerg Arndt, Aug 22 2014

A137917 a(n) is the number of unlabeled graphs on n nodes whose components are unicyclic graphs.

Original entry on oeis.org

1, 0, 0, 1, 2, 5, 14, 35, 97, 264, 733, 2034, 5728, 16101, 45595, 129327, 368093, 1049520, 2999415, 8584857, 24612114, 70652441, 203075740, 584339171, 1683151508, 4852736072, 14003298194, 40441136815, 116880901512, 338040071375, 978314772989, 2833067885748, 8208952443400

Offset: 0

Washington Bomfim, Feb 24 2008

nonn

a(n) is the number of simple unlabeled graphs on n nodes whose components have exactly one cycle. - Geoffrey Critzer, Oct 12 2012

Also the number of unlabeled simple graphs with n vertices and n edges such that it is possible to choose a different vertex from each edge. - Gus Wiseman, Jan 25 2024

			From _Gus Wiseman_, Jan 25 2024: (Start)
Representatives of the a(0) = 1 through a(5) = 5 simple graphs:
  {}  .  .  {12,13,23}  {12,13,14,23}  {12,13,14,15,23}
                        {12,13,24,34}  {12,13,14,23,25}
                                       {12,13,14,23,45}
                                       {12,13,14,25,35}
                                       {12,13,24,35,45}
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..500
F. Ruskey, Combinatorial Generation, p. 79, Eq.(4.27), 2003
Eric Weisstein's World of Mathematics, Pseudoforest.
Wikipedia, Pseudoforest.

Cf. A008483, A137918.
The connected case is A001429.
Without the choice condition we have A001434, covering A006649.
For any number of edges we have A134964, complement A140637.
The labeled version is A137916.
The version with loops is A369145, complement A368835.
The complement is counted by A369201, labeled A369143, covering A369144.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A129271 counts connected choosable simple graphs, unlabeled A005703.
Cf. A000088, A000612, A007716, A014068, A053530, A116508, A133686, A140638, A368601, A369141, A369146.

Needs["Combinatorica`"];
nn=30;s[n_,k_]:=s[n,k]=a[n+1-k]+If[n<2k,0,s[n-k,k]];a[1]=1;a[n_]:=a[n]=Sum[a[i]s[n-1,i]i,{i,1,n-1}]/(n-1);rt=Table[a[i],{i,1,nn}];c=Drop[Apply[Plus,Table[Take[CoefficientList[CycleIndex[DihedralGroup[n],s]/.Table[s[j]->Table[Sum[rt[[i]]x^(k*i),{i,1,nn}],{k,1,nn}][[j]],{j,1,nn}],x],nn],{n,3,nn}]],1];CoefficientList[Series[Product[1/(1-x^i)^c[[i]],{i,1,nn-1}],{x,0,nn}],x]   (* Geoffrey Critzer, Oct 12 2012, after code given by Robert A. Russell in A000081 *)
brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])],{p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute/@Select[Subsets[Subsets[Range[n],{2}],{n}],Select[Tuples[#],UnsameQ@@#&]!={}&]]],{n,0,5}] (* Gus Wiseman, Jan 25 2024 *)

a(n) = Sum_{1*j_1 + 2*j_2 + ... = n} (Product_{i=3..n} binomial(A001429(i) + j_i -1, j_i)). [F. Ruskey p. 79, (4.27) with n replaced by n+1, and a_i replaced by A001429(i)].

Euler transform of A001429. - Geoffrey Critzer, Oct 12 2012

Edited by Washington Bomfim, Jun 27 2012
Terms a(30) and beyond from Andrew Howroyd, May 05 2018
Offset changed to 0 by Gus Wiseman, Jan 27 2024

A026807 Triangular array T read by rows: T(n,k) = number of partitions of n in which every part is >=k, for k=1,2,...,n.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 5, 2, 1, 1, 7, 2, 1, 1, 1, 11, 4, 2, 1, 1, 1, 15, 4, 2, 1, 1, 1, 1, 22, 7, 3, 2, 1, 1, 1, 1, 30, 8, 4, 2, 1, 1, 1, 1, 1, 42, 12, 5, 3, 2, 1, 1, 1, 1, 1, 56, 14, 6, 3, 2, 1, 1, 1, 1, 1, 1, 77, 21, 9, 5, 3, 2, 1, 1, 1, 1, 1, 1, 101, 24, 10, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1, 135, 34, 13

Offset: 1

Clark Kimberling

T(n,g) is also the number of not necessarily connected 2-regular graphs with girth at least g: the part i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles. - Jason Kimberley, Feb 05 2012

			Sum_{k>=1} y^k*(-1+1/Product_{i>=0} (1-x^(k+i))) = y*x+(2*y+y^2)*x^2+(3*y+y^2+y^3)*x^3+(5*y+2*y^2+y^3+y^4)*x^4+(7*y+2*y^2+y^3+y^4+y^5)*x^5+...
Triangle starts:  - _Jason Kimberley_, Feb 05 2012
1;
2, 1;
3, 1, 1;
5, 2, 1, 1;
7, 2, 1, 1, 1;
11, 4, 2, 1, 1, 1;
15, 4, 2, 1, 1, 1, 1;
22, 7, 3, 2, 1, 1, 1, 1;
30, 8, 4, 2, 1, 1, 1, 1, 1;
42, 12, 5, 3, 2, 1, 1, 1, 1, 1;
56, 14, 6, 3, 2, 1, 1, 1, 1, 1, 1;
77, 21, 9, 5, 3, 2, 1, 1, 1, 1, 1, 1;
101, 24, 10, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1;
From _Tilman Piesk_, Feb 20 2016: (Start)
n = 12, k = 4, t = A000217(k-1) = 6
vp = A000041(n..n-t) = A000041(12..6) = (77, 56, 42, 30, 22, 15, 11)
vc = A231599(k-1, 0..t) = A231599(3, 0..6) = (1,-1,-1, 0, 1, 1,-1)
T(12, 4) = vp * transpose(vc) = 77-56-42+22+15-11 = 5
(End)

Alois P. Heinz, Rows n = 1..141, flattened
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
Tilman Piesk, Table for n = 1..30, table for n = 2..150 without values 1, illustrations of columns n = 2, 3, 4, 5, 6, 7, 8

Row sums give A046746.
Cf. A026835.
Cf. A026794.
Cf. A231599.
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: this sequence (triangle); columns of this sequence: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9). For g >= 3, girth at least g implies no loops or parallel edges. - Jason Kimberley, Feb 05 2012
Not necessarily connected 2-regular simple graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Feb 05 2012
Cf. A002260.

import Data.List (tails)
a026807 n k = a026807_tabl !! (n-1) !! (k-1)
a026807_row n = a026807_tabl !! (n-1)
a026807_tabl = map
   (\row -> map (p $ last row) $ init $ tails row) a002260_tabl
   where p 0  _ = 1
         p _ [] = 0
         p m ks'@(k:ks) = if m < k then 0 else p (m - k) ks' + p m ks
-- Reinhard Zumkeller, Dec 01 2012

T:= proc(n, k) option remember;
      `if`(k<1 or k>n, 0, `if`(n=k, 1, T(n, k+1) +T(n-k, k)))
    end:
seq(seq(T(n, k), k=1..n), n=1..14); # Alois P. Heinz, Mar 28 2012

T[n_, k_] := T[n, k] = If[ k<1 || k>n, 0, If[n == k, 1, T[n, k+1] + T[n-k, k]]]; Table [Table[ T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)

from see_there import a231599_row  # A231599
from sympy.ntheory import npartitions  # A000041
def a026807(n, k):
    if k > n:
        return 0
    elif k > n/2:
        return 1
    else:
        vc = a231599_row(k-1)
        t = len(vc)
        vp_range = range(n-t, n+1)
        vp_range = vp_range[::-1]  # reverse
        r = 0
        for i in range(0, t):
            r += vc[i] * npartitions(vp_range[i])
        return r
# Tilman Piesk, Feb 21 2016

T(n,1)=A000041(n), T(n,2)=A002865(n) for n>1, T(n,3)=A008483(n) for n>2, T(n,4)=A008484(n) for n>3.
G.f.: Sum_{k>=1} y^k*(-1+1/Product_{i>=0} (1-x^(k+i))). - Vladeta Jovovic, Jun 22 2003
T(n, k) = T(n, k+1) + T(n-k, k), T(n, k) = 1 if n/2 < k <= n. - Franklin T. Adams-Watters, Jan 24 2005; Tilman Piesk, Feb 20 2016
T(n, k) = A000041(n..n-t) * transpose(A231599(k-1, 0..t)) with t = A000217(k-1). - Tilman Piesk, Feb 20 2016
Equals A026794 * A000012 as infinite lower triangular matrices. - Gary W. Adamson, Jan 31 2008

cp's OEIS Frontend

A002865 Number of partitions of n that do not contain 1 as a part.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

Python

SageMath

Formula

A026794 Triangular array T read by rows: T(n,k) = number of partitions of n in which least part is k, 1<=k<=n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A005176 Number of regular graphs with n unlabeled nodes.

Views

Author

Keywords

References

Links

Crossrefs

Formula

Extensions

A027336 Number of partitions of n that do not contain 2 as a part.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A005638 Number of unlabeled trivalent (or cubic) graphs with 2n nodes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions

A008484 Number of partitions of n into parts >= 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Maple

Mathematica