A007717 -id:A007717 - cp's OEIS frontend

A001710 Order of alternating group A_n, or number of even permutations of n letters.

1, 1, 1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400, 239500800, 3113510400, 43589145600, 653837184000, 10461394944000, 177843714048000, 3201186852864000, 60822550204416000, 1216451004088320000, 25545471085854720000, 562000363888803840000

Offset: 0

N. J. A. Sloane

For n >= 3, a(n-1) is also the number of ways that a 3-cycle in the symmetric group S_n can be written as a product of 2 long cycles (of length n). - Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 14 2001
a(n) is the number of Hamiltonian circuit masks for an n X n adjacency matrix of an undirected graph. - Chad Brewbaker, Jan 31 2003
a(n-1) is the number of necklaces one can make with n distinct beads: n! bead permutations, divide by two to represent flipping the necklace over, divide by n to represent rotating the necklace. Related to Stirling numbers of the first kind, Stirling cycles. - Chad Brewbaker, Jan 31 2003
Number of increasing runs in all permutations of [n-1] (n>=2). Example: a(4)=12 because we have 12 increasing runs in all the permutations of [3] (shown in parentheses): (123), (13)(2), (3)(12), (2)(13), (23)(1), (3)(2)(1). - Emeric Deutsch, Aug 28 2004
Minimum permanent over all n X n (0,1)-matrices with exactly n/2 zeros. - Simone Severini, Oct 15 2004
The number of permutations of 1..n that have 2 following 1 for n >= 1 is 0, 1, 3, 12, 60, 360, 2520, 20160, ... . - Jon Perry, Sep 20 2008
Starting (1, 3, 12, 60, ...) = binomial transform of A000153: (1, 2, 7, 32, 181, ...). - Gary W. Adamson, Dec 25 2008
First column of A092582. - Mats Granvik, Feb 08 2009
The asymptotic expansion of the higher order exponential integral E(x,m=1,n=3) ~ exp(-x)/x*(1 - 3/x + 12/x^2 - 60/x^3 + 360/x^4 - 2520/x^5 + 20160/x^6 - 81440/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009
For n>1: a(n) = A173333(n,2). - Reinhard Zumkeller, Feb 19 2010
Starting (1, 3, 12, 60, ...) = eigensequence of triangle A002260, (a triangle with k terms of (1,2,3,...) in each row given k=1,2,3,...). Example: a(6) = 360, generated from (1, 2, 3, 4, 5) dot (1, 1, 3, 12, 60) = (1 + 2 + 9 + 48 + 300). - Gary W. Adamson, Aug 02 2010
For n>=2: a(n) is the number of connected 2-regular labeled graphs on (n+1) nodes (Cf. A001205). - Geoffrey Critzer, Feb 16 2011.
The Fi1 and Fi2 triangle sums of A094638 are given by the terms of this sequence (n>=1). For the definition of these triangle sums see A180662. - Johannes W. Meijer, Apr 20 2011
Also [1, 1] together with the row sums of triangle A162608. - Omar E. Pol, Mar 09 2012
a(n-1) is, for n>=2, also the number of necklaces with n beads (only C_n symmetry, no turnover) with n-1 distinct colors and signature c[.]^2 c[.]^(n-2). This means that two beads have the same color, and for n=2 the second factor is omitted. Say, cyclic(c[1]c[1]c[2]c[3]..c[n-1]), in short 1123...(n-1), taken cyclically. E.g., n=2: 11,  n=3: 112, n=4: 1123, 1132, 1213,  n=5: 11234, 11243, 11324, 11342, 11423, 11432, 12134, 12143, 13124, 13142, 14123, 14132. See the next-to-last entry in line n>=2 of the representative necklace partition array A212359. - Wolfdieter Lang, Jun 26 2012
For m >= 3, a(m-1) is the number of distinct Hamiltonian circuits in a complete simple graph with m vertices. See also A001286. - Stanislav Sykora, May 10 2014
In factorial base (A007623) these numbers have a simple pattern: 1, 1, 1, 11, 200, 2200, 30000, 330000, 4000000, 44000000, 500000000, 5500000000, 60000000000, 660000000000, 7000000000000, 77000000000000, 800000000000000, 8800000000000000, 90000000000000000, 990000000000000000, etc. See also the formula based on this observation, given below. - Antti Karttunen, Dec 19 2015
Also (by definition) the independence number of the n-transposition graph. - Eric W. Weisstein, May 21 2017
Number of permutations of n letters containing an even number of even cycles. - Michael Somos, Jul 11 2018
Equivalent to Brewbaker's and Sykora's comments, a(n - 1) is the number of undirected cycles covering n labeled vertices, hence the logarithmic transform of A002135. - Gus Wiseman, Oct 20 2018
For n >= 2 and a set of n distinct leaf labels, a(n) is the number of binary, rooted, leaf-labeled tree topologies that have a caterpillar shape (column k=1 of A306364). - Noah A Rosenberg, Feb 11 2019
Also the clique covering number of the n-Bruhat graph. - Eric W. Weisstein, Apr 19 2019
a(n) is the number of lattices of the form [s,w] in the weak order on S_n, for a fixed simple reflection s. - Bridget Tenner, Jan 16 2020
For n > 3, a(n) = p_1^e_1*...*p_m^e_m, where p_1 = 2 and e_m = 1. There exists p_1^x where x <= e_1 such that p_1^x*p_m^e_m is a primitive Zumkeller number (A180332) and p_1^e_1*p_m^e_m is a Zumkeller number (A083207). Therefore, for n > 3, a(n) = p_1^e_1*p_m^e_m*r, where r is relatively prime to p_1*p_m, is also a Zumkeller number. - Ivan N. Ianakiev, Mar 11 2020
For n>1, a(n) is the number of permutations of [n] that have 1 and 2 as cycle-mates, that is, 1 and 2 are contained in the same cycle of a cyclic representation of permutations of [n]. For example, a(4) counts the 12 permutations with 1 and 2 as cycle-mates, namely, (1 2 3 4), (1 2 4 3), (1 3 2 4), (1 3 4 2), (1 4 2 3), (1 4 3 2), (1 2 3) (4), (1 3 2) (4), (1 2 4 )(3), (1 4 2)(3), (1 2)(3 4), and (1 2)(3)(4). Since a(n+2)=row sums of A162608, our result readily follows. - Dennis P. Walsh, May 28 2020

			G.f. = 1 + x + x^2 + 3*x^3 + 12*x^4 + 60*x^5 + 360*x^6 + 2520*x^7 + ...

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 87-8, 20. (a), c_n^e(t=1).
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).

N. J. A. Sloane, Table of n, a(n) for n = 0..100
Somaya Barati, Beáta Bényi, Abbas Jafarzadeh, and Daniel Yaqubi, Mixed restricted Stirling numbers, arXiv:1812.02955 [math.CO], 2018.
Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.
Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
Jonathan Beagley and Lara Pudwell, Colorful Tilings and Permutations, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.
Olivier Bodini, Antoine Genitrini, and Mehdi Naima, Ranked Schröder Trees, arXiv:1808.08376 [cs.DS], 2018.
Olivier Bodini, Antoine Genitrini, Cécile Mailler, and Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), Article 00.1.5.
Camille Combe and Samuele Giraudo, Cliff operads: a hierarchy of operads on words, arXiv:2106.14552 [math.CO], 2021.
Mareike Fischer, Extremal Values of the Sackin Tree Balance Index, Ann. Comb. (2021) Vol. 25, 515-541, Remark 7.
Hannah Golab, Pattern avoidance in Cayley permutations, Master's Thesis, Northern Arizona Univ. (2024). See p. 36.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 262.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Shirali Kadyrov and Farukh Mashurov, Generalized continued fraction expansions for Pi and E, arXiv:1912.03214 [math.NT], 2019.
Chanchal Kumar and Amit Roy, Integer Sequences and Monomial Ideals, arXiv:2003.10098 [math.CO], 2020.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
Xah Lee, Combinatorics: Loop in n points.
D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
S-Z Song, S-G Hwang, S-H Rim, and G-S Cheon, Extremes of permanents of (0,1)-matrices, Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002). Linear Algebra Appl. 373 (2003), 197-210.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
B. E. Tenner, Interval structures in the Bruhat and weak orders, arXiv:2001.05011 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Alternating Group.
Eric Weisstein's World of Mathematics, Bruhat Graph.
Eric Weisstein's World of Mathematics, Circular Permutation.
Eric Weisstein's World of Mathematics, Clique Covering Number.
Eric Weisstein's World of Mathematics, Even Permutation.
Eric Weisstein's World of Mathematics, Hamiltonian Cycle.
Eric Weisstein's World of Mathematics, Independence Number.
Eric Weisstein's World of Mathematics, Odd Permutation.
Eric Weisstein's World of Mathematics, Transposition Graph.
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 7.
Index entries for sequences related to factorial base representation.
Index entries for sequences related to factorial numbers.
Index entries for sequences related to groups.
Index to divisibility sequences.

Cf. A000142, A000153, A000255, A001147, A001286, A001720, A002135, A002260, A007623, A007717, A049444, A049459, A093468, A094587, A094638, A138533, A153880, A173333, A213936, A215771, A319225, A319226, A320655.
a(n+1)= A046089(n, 1), n >= 1 (first column of triangle), A161739 (q(n) sequence).
Bisections are A002674 and A085990 (essentially).
Row 3 of A265609 (essentially).
Row sums of A307429.

[1] cat [Order(AlternatingGroup(n)): n in [1..20]]; // Arkadiusz Wesolowski, May 17 2014

seq(mul(k, k=3..n), n=0..20); # Zerinvary Lajos, Sep 14 2007

a[n_]:= If[n > 2, n!/2, 1]; Array[a, 21, 0]
a[n_]:= If[n<3, 1, n*a[n-1]]; Array[a, 21, 0]; (* Robert G. Wilson v, Apr 16 2011 *)
a[ n_]:= If[n<0, 0, n! SeriesCoefficient[(2-x^2)/(2-2x), {x, 0, n}]]; (* Michael Somos, May 22 2014 *)
a[ n_]:= If[n<0, 0, n! SeriesCoefficient[1 +Sinh[-Log[1-x]], {x, 0, n}]]; (* Michael Somos, May 22 2014 *)
Numerator[Range[0, 20]!/2] (* Eric W. Weisstein, May 21 2017 *)
Table[GroupOrder[AlternatingGroup[n]], {n, 0, 20}] (* Eric W. Weisstein, May 21 2017 *)

PARI
```
{a(n) = if( n<2, n>=0, n!/2)};
```

a(n)=polcoeff(1+x*sum(m=0,n,m^m*x^m/(1+m*x+x*O(x^n))^m),n) \\ Paul D. Hanna

A001710=n->n!\2+(n<2) \\ M. F. Hasler, Dec 01 2013

from math import factorial
def A001710(n): return factorial(n)>>1 if n > 1 else 1 # Chai Wah Wu, Feb 14 2023

def A001710(n): return (factorial(n) +int(n<2))//2
[A001710(n) for n in range(31)] # G. C. Greubel, Sep 28 2024

;; Using memoization-macro definec for which an implementation can be found in http://oeis.org/wiki/Memoization
(definec (A001710 n) (cond ((<= n 2) 1) (else (* n (A001710 (- n 1))))))
;; Antti Karttunen, Dec 19 2015

a(n) = numerator(n!/2) and A141044(n) = denominator(n!/2).
D-finite with recurrence: a(0) = a(1) = a(2) = 1; a(n) = n*a(n-1) for n>2. - Chad Brewbaker, Jan 31 2003 [Corrected by N. J. A. Sloane, Jul 25 2008]
a(0) = 0, a(1) = 1; a(n) = Sum_{k=1..n-1} k*a(k). - Amarnath Murthy, Oct 29 2002
Stirling transform of a(n+1) = [1, 3, 12, 160, ...] is A083410(n) = [1, 4, 22, 154, ...]. - Michael Somos, Mar 04 2004
First Eulerian transform of A000027. See A000142 for definition of FET. - Ross La Haye, Feb 14 2005
From Paul Barry, Apr 18 2005: (Start)
a(n) = 0^n + Sum_{k=0..n} (-1)^(n-k-1)*T(n-1, k)*cos(Pi*(n-k-1)/2)^2.
T(n,k) = abs(A008276(n, k)). (End)
E.g.f.: (2 - x^2)/(2 - 2*x).
E.g.f. of a(n+2), n>=0, is 1/(1-x)^3.
E.g.f.: 1 + sinh(log(1/(1-x))). - Geoffrey Critzer, Dec 12 2010
a(n+1) = (-1)^n * A136656(n,1), n>=1.
a(n) = n!/2 for n>=2 (proof from the e.g.f). - Wolfdieter Lang, Apr 30 2010
a(n) = (n-2)! * t(n-1), n>1, where t(n) is the n-th triangular number (A000217). - Gary Detlefs, May 21 2010
a(n) = ( A000254(n) - 2* A001711(n-3) )/3, n>2. - Gary Detlefs, May 24 2010
O.g.f.: 1 + x*Sum_{n>=0} n^n*x^n/(1 + n*x)^n. - Paul D. Hanna, Sep 13 2011
a(n) = if n < 2 then 1, otherwise Pochhammer(n,n)/binomial(2*n,n). - Peter Luschny, Nov 07 2011
a(n) = Sum_{k=0..floor(n/2)} s(n,n-2*k) where s(n,k) are Stirling number of the first kind, A048994. - Mircea Merca, Apr 07 2012
a(n-1), n>=3, is M_1([2,1^(n-2)])/n = (n-1)!/2, with the M_1 multinomial numbers for the given n-1 part partition of n. See the second to last entry in line n>=3 of A036038, and the above necklace comment by W. Lang. - Wolfdieter Lang, Jun 26 2012
G.f.: A(x) = 1 + x + x^2/(G(0)-2*x) where G(k) = 1 - (k+1)*x/(1 - x*(k+3)/G(k+1)); (continued fraction). - Sergei N. Gladkovskii, Dec 26 2012.
G.f.: 1 + x + (Q(0)-1)*x^2/(2*(sqrt(x)+x)), where Q(k) = 1 + (k+2)*sqrt(x)/(1 - sqrt(x)/(sqrt(x) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
G.f.: 1 + x + (x*Q(x)-x^2)/(2*(sqrt(x)+x)), where Q(x) = Sum_{n>=0} (n+1)!*x^n*sqrt(x)*(sqrt(x) + x*(n+2)). - Sergei N. Gladkovskii, May 15 2013
G.f.: 1 + x/2 + (Q(0)-1)*x/(2*(sqrt(x)+x)), where Q(k) = 1 + (k+1)*sqrt(x)/(1 - sqrt(x)/(sqrt(x) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
G.f.: 1 + x + x^2*G(0)/2, where G(k) = 1 + 1/(1 - x/(x + 1/(k+3)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
G.f.: 1+x + x^2*W(0), where W(k) = 1 - x*(k+3)/( x*(k+3) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013
From Antti Karttunen, Dec 19 2015: (Start)
a(0)=a(1)=1; after which, for even n: a(n) = (n/2) * (n-1)!, and for odd n: a(n) = (n-1)/2 * ((n-1)! + (n-2)!). [The formula was empirically found after viewing these numbers in factorial base, A007623, and is easily proved by considering formulas from Lang (Apr 30 2010) and Detlefs (May 21 2010) shown above.]
For n >= 1, a(2*n+1) = a(2*n) + A153880(a(2*n)). [Follows from above.] (End)
Inverse Stirling transform of a(n) is (-1)^(n-1)*A009566(n). - Anton Zakharov, Aug 07 2016
a(n) ~ sqrt(Pi/2)*n^(n+1/2)/exp(n). - Ilya Gutkovskiy, Aug 07 2016
a(n) = A006595(n-1)*n/A000124(n) for n>=2. - Anton Zakharov, Aug 23 2016
a(n) = A001563(n-1) - A001286(n-1) for n>=2. - Anton Zakharov, Sep 23 2016
From Peter Bala, May 24 2017: (Start)
The o.g.f. A(x) satisfies the Riccati equation x^2*A'(x) + (x - 1)*A(x) + 1 - x^2 = 0.
G.f.: A(x) = 1 + x + x^2/(1 - 3*x/(1 - x/(1 - 4*x/(1 - 2*x/(1 - 5*x/(1 - 3*x/(1  - ... - (n + 2)*x/(1 - n*x/(1 - ... ))))))))) (apply Stokes, 1982).
A(x) = 1 + x + x^2/(1 - 2*x - x/(1 - 3*x/(1 - 2*x/(1 - 4*x/(1 - 3*x/(1 - 5*x/(1 - ... - n*x/(1 - (n+2)*x/(1 - ... ))))))))). (End)
H(x) = (1 - (1 + x)^(-2)) / 2 = x - 3*x^2/2! + 12*x^3/3! - ..., an e.g.f. for the signed sequence here (n!/2!), ignoring the first two terms, is the compositional inverse of G(x) = (1 - 2*x)^(-1/2) - 1 = x + 3*x^2/2! + 15*x^3/3! + ..., an e.g.f. for A001147. Cf. A094638. H(x) is the e.g.f. for the sequence (-1)^m * m!/2 for m = 2,3,4,... . Cf. A001715 for n!/3! and A001720 for n!/4!. Cf. columns of A094587, A173333, and A213936 and rows of A138533. - Tom Copeland, Dec 27 2019
From Amiram Eldar, Jan 08 2023: (Start)
Sum_{n>=0} 1/a(n) = 2*(e-1).
Sum_{n>=0} (-1)^n/a(n) = 2/e. (End)

More terms from Larry Reeves (larryr(AT)acm.org), Aug 20 2001

Further terms from Simone Severini, Oct 15 2004

A316980 Number of non-isomorphic strict multiset partitions of weight n.

Original entry on oeis.org

1, 1, 3, 8, 23, 63, 197, 588, 1892, 6140, 20734, 71472, 254090, 923900, 3446572, 13149295, 51316445, 204556612, 832467052, 3455533022, 14621598811, 63023667027, 276559371189, 1234802595648, 5606647482646, 25875459311317, 121324797470067, 577692044073205

Offset: 0

Gus Wiseman, Jul 18 2018

nonn

Also the number of nonnegative integer n X n matrices with sum of elements equal to n, under row and column permutations, with no equal rows (or alternatively, with no equal columns).

Also the number of non-isomorphic multiset partitions of weight n with no equivalent vertices. In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second.

			Non-isomorphic representatives of the a(3) = 8 multiset partitions with no equivalent vertices (first column) and with no equal blocks (second column):
      (111) <-> (111)
      (122) <-> (1)(11)
    (1)(11) <-> (122)
    (1)(22) <-> (1)(22)
    (2)(12) <-> (2)(12)
  (1)(1)(1) <-> (123)
  (1)(2)(2) <-> (1)(23)
  (1)(2)(3) <-> (1)(2)(3)

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A000009, A001055, A007716, A007717, A020555, A045778, A130091.
Cf. A316974, A316978, A316979, A316981, A316983.
Cf. A049311, A059201, A319558, A319560, A319567.

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(p=sum(t=1, n, subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*polcoef(exp(p-subst(p,x,x^2)), n)); s/n!)} \\ Andrew Howroyd, Jan 21 2023

Euler transform of A319557. - Gus Wiseman, Sep 23 2018

a(7)-a(10) from Gus Wiseman, Sep 23 2018

Terms a(11) and beyond from Andrew Howroyd, Jan 19 2023

A316983 Number of non-isomorphic self-dual multiset partitions of weight n.

Original entry on oeis.org

1, 1, 2, 4, 9, 17, 36, 72, 155, 319, 677, 1429, 3094, 6648, 14518, 31796, 70491, 156818, 352371, 795952, 1813580, 4155367, 9594425, 22283566, 52122379, 122631874, 290432439, 691831161, 1658270316, 3997272089, 9692519896, 23631827354, 57943821449, 142834652193

Offset: 0

Gus Wiseman, Jul 18 2018

nonn

Also the number of nonnegative integer square symmetric matrices with sum of elements equal to n, under row and column permutations.

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity.

			Non-isomorphic representatives of the a(4) = 9 self-dual multiset partitions:
  (1111),
  (1)(222), (2)(122), (11)(22), (12)(12),
  (1)(1)(23), (1)(2)(33), (1)(3)(23),
  (1)(2)(3)(4).
The a(4) = 9 square symmetric matrices:
. [4]
.
. [3 0]  [2 0]  [2 1]  [1 1]
. [0 1]  [0 2]  [1 0]  [1 1]
.
. [2 0 0]  [1 1 0]  [0 1 1]
. [0 1 0]  [1 0 0]  [1 0 0]
. [0 0 1]  [0 0 1]  [1 0 0]
.
. [1 0 0 0]
. [0 1 0 0]
. [0 0 1 0]
. [0 0 0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..50

Row sums of A320796.
Main diagonal of A318805.
Cf. A000009, A001055, A007716, A007717, A020555, A045778.
Cf. A316974, A316978, A316979, A316980, A316981.

vector(25, n, n--; T(n,n)) \\ T(n,k) defined in A318805. - Andrew Howroyd, Jan 16 2024

Terms a(9) and beyond from Andrew Howroyd, Sep 03 2018

A000569 Number of graphical partitions of 2n.

Original entry on oeis.org

1, 2, 5, 9, 17, 31, 54, 90, 151, 244, 387, 607, 933, 1420, 2136, 3173, 4657, 6799, 9803, 14048, 19956, 28179, 39467, 54996, 76104, 104802, 143481, 195485, 264941, 357635, 480408, 642723, 856398, 1136715, 1503172, 1980785

Offset: 1

N. J. A. Sloane

A partition of n is a sequence p_1, ..., p_k for some k with p_1 >= p_2 >= ... >= p_k and p_1+...+p_k=n. A partition is graphical if it is the degree sequence of a simple graph (this requires that n be even). Some authors set a(0)=1 by convention.

			a(2)=2: the graphical partitions of 4 are 2+1+1 and 1+1+1+1, corresponding to the degree sequences of the graphs V and ||.
From _Gus Wiseman_, Oct 26 2018: (Start)
The a(1) = 1 through a(5) = 17 graphical partitions:
  (11)  (211)   (222)     (2222)      (3322)
        (1111)  (2211)    (3221)      (22222)
                (3111)    (22211)     (32221)
                (21111)   (32111)     (33211)
                (111111)  (41111)     (42211)
                          (221111)    (222211)
                          (311111)    (322111)
                          (2111111)   (331111)
                          (11111111)  (421111)
                                      (511111)
                                      (2221111)
                                      (3211111)
                                      (4111111)
                                      (22111111)
                                      (31111111)
                                      (211111111)
                                      (1111111111)
(End)

T. D. Noe, Table of n, a(n) for n = 1..860. [Terms 1 through 110 were computed by Tiffany M. Barnes and Carla D. Savage; terms 111 through 585 were computed by Axel Kohnert; terms 586 to 860 by Wang Kai, Jun 05 2016; a typo of a(547) in Number of Graphical Partitions is corrected by Wang Kai, Aug 03 2016]
Tiffany M. Barnes and Carla D. Savage, Efficient generation of graphical partitions Discrete Appl. Math. 78 (1997), no. 1-3, 17-26.
T. M. Barnes and C. D. Savage, A recurrence for counting graphical partitions, Electronic J. Combinatorics, 2 (1995).
K. Blum, Bounds on the Number of Graphical Partitions, arXiv:2103.03196 [math.CO], 2021. See Table on p. 7.
P. Erdős and T. Gallai, Graphs with a given degree of vertices, Mat. Lapok, 11 (1960), 264-274.
P. Erdős and L. B. Richmond, On graphical partitions Combinatorica 13 (1993), no. 1, 57-63.
Axel Kohnert, Dominance Order and Graphical Partitions, Electronic J. Combinatorics, 11 (2004).
Gerard Sierksma and Han Hoogeveen, Seven criteria for integer sequences being graphic, J. Graph Theory 15 (1991), no. 2, 223-231.
Eric Weisstein's World of Mathematics, Graphical partition.
Gus Wiseman, Simple graphs realizing each of the a(6) = 31 graphical partitions of 12.
Index entries for sequences related to graphical partitions

Cf. A000070, A000219, A004250, A004251, A007717, A025065, A029889, A095268, A096373, A147878, A209816, A320911, A320921, A320922.

<< MathWorld`Graphs`
Table[Count[RealizeDegreeSequence /@ Partitions[n], _Graph], {n, 2, 20, 2}]
(* second program *)
prptns[m_]:=Union[Sort/@If[Length[m]==0,{{}},Join@@Table[Prepend[#,m[[ipr]]]&/@prptns[Delete[m,List/@ipr]],{ipr,Select[Prepend[{#},1]&/@Select[Range[2,Length[m]],m[[#]]>m[[#-1]]&],UnsameQ@@m[[#]]&]}]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[2*n],Select[prptns[#],UnsameQ@@#&]!={}&]],{n,6}] (* Gus Wiseman, Oct 26 2018 *)

A320655 Number of factorizations of n into semiprimes. Number of multiset partitions of the multiset of prime factors of n, into pairs.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 2, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 2, 1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 1, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0

Offset: 1

Gus Wiseman, Oct 18 2018

nonn

The characteristic function of nonzero terms is A065043. - R. J. Mathar, Jan 18 2021

			The a(900) = 5 factorizations into semiprimes:
  900 = (4*9*25)
  900 = (4*15*15)
  900 = (6*6*25)
  900 = (6*10*15)
  900 = (9*10*10)
The a(900) = 5 multiset partitions into pairs:
  {{1,1},{2,2},{3,3}}
  {{1,1},{2,3},{2,3}}
  {{1,2},{1,2},{3,3}}
  {{1,2},{1,3},{2,3}}
  {{2,2},{1,3},{1,3}}

Antti Karttunen, Table of n, a(n) for n = 1..65537

The positions of zeros are A026424.

Cf. A001055, A001222, A001358, A007716, A007717, A056239, A112798, A318871, A318953, A320462, A320656, A320658, A320659.

semfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[semfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];
Table[Length[semfacs[n]],{n,100}]

A320655(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((2==bigomega(d)&&(d<=m)), s += A320655(n/d, d))); (s)); \\ Antti Karttunen, Dec 06 2020

Data section extended up to 105 terms by Antti Karttunen, Dec 06 2020

A320656 Number of factorizations of n into squarefree semiprimes. Number of multiset partitions of the multiset of prime factors of n, into strict pairs.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0

Offset: 1

Gus Wiseman, Oct 18 2018

nonn

			The a(4620) = 6 factorizations into squarefree semiprimes:
  4620 = (6*10*77)
  4620 = (6*14*55)
  4620 = (6*22*35)
  4620 = (10*14*33)
  4620 = (10*21*22)
  4620 = (14*15*22)
The a(4620) = 6 multiset partitions into strict pairs:
  {{1,2},{1,3},{4,5}}
  {{1,2},{1,4},{3,5}}
  {{1,2},{1,5},{3,4}}
  {{1,3},{1,4},{2,5}}
  {{1,3},{2,4},{1,5}}
  {{1,4},{2,3},{1,5}}
The a(69300) = 10 factorizations into squarefree semiprimes:
  69300 = (6*6*35*55)
  69300 = (6*10*15*77)
  69300 = (6*10*21*55)
  69300 = (6*10*33*35)
  69300 = (6*14*15*55)
  69300 = (6*15*22*35)
  69300 = (10*10*21*33)
  69300 = (10*14*15*33)
  69300 = (10*15*21*22)
  69300 = (14*15*15*22)
The a(69300) = 10 multiset partitions into strict pairs:
  {{1,2},{1,2},{3,4},{3,5}}
  {{1,2},{1,3},{2,3},{4,5}}
  {{1,2},{1,3},{2,4},{3,5}}
  {{1,2},{1,3},{2,5},{3,4}}
  {{1,2},{1,4},{2,3},{3,5}}
  {{1,2},{2,3},{1,5},{3,4}}
  {{1,3},{1,3},{2,4},{2,5}}
  {{1,3},{1,4},{2,3},{2,5}}
  {{1,3},{2,3},{2,4},{1,5}}
  {{1,4},{2,3},{2,3},{1,5}}.
The a(210) = 3 factorizations into squarefree semiprimes: 210 = (6*35) = (10*21) = (14*15). - _Antti Karttunen_, Nov 02 2022

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A001055, A002110, A006881, A079275, A007716, A007717, A045778, A123023, A318871, A318953, A320462, A320655, A320658, A320659.

bepfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[bepfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
Table[Length[bepfacs[n]],{n,100}]

A320656(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&issquarefree(d)&&2==bigomega(d), s += A320656(n/d, d))); (s)); \\ Antti Karttunen, Nov 02 2022

a(A002110(n)) = A123023(n). - Antti Karttunen, Nov 02 2022

Data section extended up to a(120) and the secondary offset added by Antti Karttunen, Nov 02 2022

A320911 Numbers with an even number of prime factors (counted with multiplicity) that can be factored into squarefree semiprimes.

Original entry on oeis.org

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 126, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

A squarefree semiprime (A006881) is a product of any two distinct primes.

Also numbers with an even number x of prime factors, whose prime multiplicities do not exceed x/2.

			360 is in the sequence because it can be factored into squarefree semiprimes as (6*6*10).
4620 is in the sequence, and can be factored into squarefree semiprimes in 6 ways: (6*10*77), (6*14*55), (6*22*35), (10*14*33), (10*21*22), (14*15*22).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001055, A001222, A001358, A005117, A006881, A007717, A028260, A320655, A320656, A320891, A320892, A320893, A320894, A320912.

sqfsemfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfsemfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
Select[Range[100],And[EvenQ[PrimeOmega[#]],sqfsemfacs[#]!={}]&]

A050535 Number of loopless multigraphs on infinite set of nodes with n edges.

Original entry on oeis.org

1, 1, 3, 8, 23, 66, 212, 686, 2389, 8682, 33160, 132277, 550835, 2384411, 10709827, 49782637, 238998910, 1182772364, 6023860266, 31525780044, 169316000494, 932078457785, 5253664040426, 30290320077851, 178480713438362, 1073918172017297

Offset: 0

Vladeta Jovovic, Dec 29 1999

nonn

Also, a(n) is the number of n-rowed binary matrices with all row sums equal to 2, up to row and column permutation (see Jovovic's formula). Also, a(n) is the limit of A192517(m,n) as m grows. - Max Alekseyev, Oct 18 2017
Row sums of the triangle defined by the Multiset Transformation of A076864,
;
1;
2 1;
5 2 1;
12 8 2 1;
33 22 8 2 1;
103 72 26 8 2 1;
333 229 87 26 8 2 1;
1183 782 295 92 26 8 2 1;
4442 2760 1036 315 92 26 8 2 1;
17576 10270 3735 1129 321 92 26 8 2 1;
72810 39770 13976 4117 1154 321 92 26 8 2 1;
314595 160713 54132 15547 4237 1161 321 92 26 8 2 1;
- R. J. Mathar, Jul 18 2017
Also the number of non-isomorphic set multipartitions (multisets of sets) of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018

			From _Gus Wiseman_, Jul 18 2018: (Start)
Non-isomorphic representatives of the a(3) = 8 set multipartitions of {1, 1, 2, 2, 3, 3}:
  (123)(123)
  (1)(23)(123)
  (12)(13)(23)
  (1)(1)(23)(23)
  (1)(2)(3)(123)
  (1)(2)(13)(23)
  (1)(1)(2)(3)(23)
  (1)(1)(2)(2)(3)(3)
(End)

Frank Harary and Edgar M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, Eq. (4.1.18).

Andrew Howroyd, Table of n, a(n) for n = 0..50
George Barnes, Sanjaye Ramgoolam, and Michael Stephanou, Permutation invariant Gaussian matrix models for financial correlation matrices, arXiv:2306.04569 [q-fin.ST], 2023.
Frank Harary, The number of linear, directed, rooted, and connected graphs, Trans. Am. Math. Soc. 78 (1955) 445-463, eq. (24).
Vladeta Jovovic, Number of m-rowed binary matrices with all row sums equal to n, up to row and column permutation
Patrick T. Komiske, Eric M. Metodiev, and Jesse Thaler, Energy flow polynomials: A complete linear basis for jet substructure, arXiv:1712.07124 [hep-ph], 2017.
Tsuyoshi Miezaki, Akihiro Munemasa, Yusaku Nishimura, Tadashi Sakuma, and Shuhei Tsujie, Universal graph series, chromatic functions, and their index theory, arXiv:2403.09985 [math.CO], 2024. See p. 23.

Cf. A001399, A003082, A014395, A014396, A014397, A014398.
Cf. A058389, A050913, A058783, A058390, A058784, A058785, A058391, A058392, A001501, A058528.
Cf. A007716, A007717, A020555, A050535, A076864 (inverse Euler transf.), A076867 (Euler transform) A094574, A316974.

seq[n_] := G[2n, x+O[x]^n, {}] // CoefficientList[#, x]&;
seq[15] (* Jean-François Alcover, Dec 02 2020, using Andrew Howroyd's code for G in A339065 *)

a(n) = A192517(2*n,n) = A192517(m,n) for any m>=2*n. - Max Alekseyev, Oct 18 2017

Euler transform of A076864. - Andrew Howroyd, Oct 23 2019

More terms from Sean A. Irvine, Oct 02 2011

A096373 Number of partitions of n such that the least part occurs exactly twice.

Original entry on oeis.org

0, 1, 0, 2, 1, 3, 3, 6, 5, 11, 11, 17, 20, 30, 33, 49, 56, 77, 92, 122, 143, 190, 225, 287, 344, 435, 516, 648, 770, 951, 1134, 1388, 1646, 2007, 2376, 2868, 3395, 4078, 4807, 5749, 6764, 8042, 9449, 11187, 13101, 15463, 18070, 21236, 24772, 29021, 33764

Offset: 1

Vladeta Jovovic, Jul 19 2004

Also number of partitions of n such that the difference between the two largest distinct parts is 2 (it is assumed that 0 is a part in each partition). Example: a(6)=3 because we have [4,2], [3,1,1,1] and [2,2,2]. - Emeric Deutsch, Apr 08 2006
Number of partitions p of n+2 such that min(p) + (number of parts of p) is a part of p. - Clark Kimberling, Feb 27 2014
Number of partitions of n+1 such that the two smallest parts differ by one. - Giovanni Resta, Mar 07 2014
Also the number of integer partitions of n with an even number of parts that cannot be grouped into pairs of distinct parts. These are also integer partitions of n with an even number of parts whose greatest multiplicity is greater than half the number of parts. - Gus Wiseman, Oct 26 2018

			a(6)=3 because we have [4,1,1], [3,3] and [2,2,1,1].
G.f. = x^2 + 2*x^4 + x^5 + 3*x^6 + 3*x^7 + 6*x^8 + 5*x^9 + 11*x^10 + 11*x^11 + ...
From _Gus Wiseman_, Oct 26 2018: (Start)
The a(2) = 1 through a(10) = 11 partitions where the least part occurs exactly twice (zero terms not shown):
  (11)  (22)   (311)  (33)    (322)   (44)     (522)    (55)
        (211)         (411)   (511)   (422)    (711)    (433)
                      (2211)  (3211)  (611)    (4311)   (622)
                                      (3311)   (5211)   (811)
                                      (4211)   (32211)  (3322)
                                      (22211)           (4411)
                                                        (5311)
                                                        (6211)
                                                        (33211)
                                                        (42211)
                                                        (222211)
The a(2) = 1 through a(10) = 11 partitions that cannot be grouped into pairs of distinct parts (zero terms not shown):
  (11) (22)   (2111) (33)     (2221)   (44)       (3222)     (55)
       (1111)        (3111)   (4111)   (2222)     (6111)     (3331)
                     (111111) (211111) (5111)     (321111)   (4222)
                                       (221111)   (411111)   (7111)
                                       (311111)   (21111111) (222211)
                                       (11111111)            (331111)
                                                             (421111)
                                                             (511111)
                                                             (22111111)
                                                             (31111111)
                                                             (1111111111)
(End)

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A002865, A097091, A097092, A097093.
Cf. A117989.
Cf. A000070, A000569, A007717, A025065, A141268, A209816, A261049, A320328, A320891, A320921, A320922.

g:=sum(x^(2*k)/product(1-x^j,j=k+1..80),k=1..70): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=1..51); # Emeric Deutsch, Apr 08 2006

(* do first *) Needs["DiscreteMath`Combinatorica`"] (* then *) f[n_] := Block[{p = Partitions[n], l = PartitionsP[n], c = 0, k = 1}, While[k < l + 1, q = PadLeft[ p[[k]], 3]; If[ q[[1]] != q[[3]] && q[[2]] == q[[3]], c++ ]; k++ ]; c]; Table[ f[n], {n, 51}] (* Robert G. Wilson v, Jul 23 2004 *)
Table[Count[IntegerPartitions[n+2], p_ /; MemberQ[p, Length[p] + Min[p]]], {n, 50}] (* Clark Kimberling, Feb 27 2014 *)
p[n_, m_] := If[m == n, 1, If[m > n, 0, p[n, m] = Sum[p[n-m, k], {k, m, n}]]];
a[n_] := Sum[p[n+1-k, k+1], {k, n/2}]; Array[a, 100] (* Giovanni Resta, Mar 07 2014 *)

{q=sum(m=1,100,x^(2*m)/prod(i=m+1,100,1-x^i,1+O(x^60)),1+O(x^60));for(n=1,51,print1(polcoeff(q,n),","))} \\ Klaus Brockhaus, Jul 21 2004

{a(n) = if( n<0, 0, polcoeff( ( 1 - (1 - x - x^2) / eta(x + x^4 * O(x^n)) ) * (1 - x) / x^3, n))} /* Michael Somos, Feb 28 2014 */

G.f.: Sum_{m>0} (x^(2*m) / Product_{i>m} (1-x^i)). More generally, g.f. for number of partitions of n such that the least part occurs exactly k times is Sum_{m>0} (x^(k*m)/Product_{i>m} (1-x^i)).
G.f.: Sum_{k>=1} (x^(2*k-2)*(1-x^(k-1))/Product_{j=1..k} (1-x^j)). - Emeric Deutsch, Apr 08 2006
a(n) = -p(n+3)+2*p(n+2)-p(n), p(n)=A000041(n). - Mircea Merca, Jul 10 2013
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi / (12*sqrt(2)*n^(3/2)). - Vaclav Kotesovec, Jun 02 2018

Edited and extended by Robert G. Wilson v and Klaus Brockhaus, Jul 21 2004

cp's OEIS Frontend

A068593 Erroneous version of A007717.

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A001710 Order of alternating group A_n, or number of even permutations of n letters.

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A316980 Number of non-isomorphic strict multiset partitions of weight n.

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A316983 Number of non-isomorphic self-dual multiset partitions of weight n.

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A000569 Number of graphical partitions of 2n.

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A320655 Number of factorizations of n into semiprimes. Number of multiset partitions of the multiset of prime factors of n, into pairs.

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A320656 Number of factorizations of n into squarefree semiprimes. Number of multiset partitions of the multiset of prime factors of n, into strict pairs.

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