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A000712
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Generating function = Product_{m>=1} 1/(1 - x^m)^2; a(n) = number of partitions of n into parts of 2 kinds.
(Formerly M1376 N0536)
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185
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1, 2, 5, 10, 20, 36, 65, 110, 185, 300, 481, 752, 1165, 1770, 2665, 3956, 5822, 8470, 12230, 17490, 24842, 35002, 49010, 68150, 94235, 129512, 177087, 240840, 326015, 439190, 589128, 786814, 1046705, 1386930, 1831065, 2408658, 3157789, 4126070, 5374390
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OFFSET
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0,2
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COMMENTS
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For n >= 1, a(n) is also the number of conjugacy classes in the automorphism group of the n-dimensional hypercube. This automorphism group is the wreath product of the cyclic group C_2 and the symmetric group S_n, its order is in sequence A000165. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 04 2001
Also, number of noncongruent matrices in GL_n(Z): each Jordan block can only have +1 or -1 on the diagonal. - Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 15 2004
a(n) = Sum (k(1)+1)*(k(2)+1)*...*(k(n)+1), where the sum is taken over all (k(1),k(2),...,k(n)) such that k(1)+2*k(2)+...+n*k(n) = n, k(i)>=0, i=1..n, cf. A104510, A077285. - Vladeta Jovovic, Apr 21 2005
Number of one-to-one partial endofunctions on n unlabeled points. Connected components are either cycles or "lines", hence two for each size. - Franklin T. Adams-Watters, Dec 28 2006
Equals A000716: (1, 3, 9, 22, 561, 108, ...) convolved with A010815. A000716 = the number of partitions of n into parts of 3 kinds = the Euler transform of [3,3,3,...]. - Gary W. Adamson, Oct 26 2008
Paraphrasing the g.f.: 1 + 2x + 5x^2 + ... = s(x) * s(x^2) * s(x^3) * s(x^4) * ...; where s(x) = 1 + 2x + 3x^2 + 4x^3 + ... is (up to a factor x) the g.f. of A000027. - Gary W. Adamson, Apr 01 2010
Also equals number of partitions of 2n in which the odd parts appear as many times in even as in odd positions. - Wouter Meeussen, Apr 17 2013
Also number of ordered pairs (R,S) with R a partition of r, S a partition of s, and r+s=n; see example. This corresponds to the formula a(n) = sum(r+s==n, p(r)*p(s) ) = Sum_{k=0..n} p(k)*p(n-k). - Joerg Arndt, Apr 29 2013
Also the number of all multi-graphs with exactly n-edges and with vertex degrees 1 or 2. - Ebrahim Ghorbani, Dec 02 2013
If one decomposes k-permutations into cycles and so-called paths, the number of different type of decompositions equals to a(k); see the paper by Chen, Ghorbani, and Wong. - Ebrahim Ghorbani, Dec 02 2013
Let T(n,k) be the number of partitions of n having parts 1 through k of two kinds, with T(n,0) = A000041(n), the number of partitions of n. Then a(n) = T(n,0) + T(n-1,1) + T(n-2,2) + T(n-3,3) + ... - Gregory L. Simay, May 18 2019
Also the number of orbits of projections in the partition monoid P_n under conjugation by permutations. - James East, Jul 21 2020
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REFERENCES
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H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
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).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see Proposition 2.5.2 on page 78.
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LINKS
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M. De Salvo, D. Fasino, D. Freni, and G. Lo Faro, Fully simple semihypergroups, transitive digraphs, and sequence A000712, Journal of Algebra, Volume 415, 1 October 2014, pp. 65-87.
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FORMULA
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a(n) = Sum_{k=0..n} p(k)*p(n-k), where p(n) = A000041(n).
Euler transform of period 1 sequence [ 2, 2, 2, ...]. - Michael Somos, Jul 22 2003
a(0) = 1, a(n) = (1/n)*Sum_{k=0..n-1} 2*a(k)*sigma_1(n-k)). - Joerg Arndt, Feb 05 2011
a(n) ~ (1/12)*3^(1/4)*n^(-5/4)*exp((2/3)*sqrt(3)*Pi*sqrt(n)). - Joe Keane (jgk(AT)jgk.org), Sep 13 2002
More precise asymptotics: a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*3^(3/4)*n^(5/4)) * (1 - (Pi/(12*sqrt(3)) + 15*sqrt(3)/(16*Pi)) / sqrt(n) + (Pi^2/864 + 315/(512*Pi^2) + 35/192)/n). - Vaclav Kotesovec, Jan 22 2017
a(n) is odd iff n = 2*m and p(m) is odd.
a(n) = (2/n)*Sum_{k = 0..n} k*p(k)*p(n-k) for n >= 1.
Conjecture: : a(n) is divisible by 5 when n is congruent to 2, 3 or 4 modulo 5. (End)
Conjecture is proved in Hammond and Lewis. - Yen-chi R. Lin, Jun 24 2024
With the convention that a(n) = 0 for n < 0 we have the recurrence a(n) = g(n) + Sum_{k >= 1} (-1)^(k+1)*(2*k + 1)*a(n - k*(k + 1)/2), where g(n) = (-1)^m if n = m*(3*m - 1)/2 is a generalized pentagonal number (A001318) else g(n) = 0. For example, n = 7 = -2*(3*(-2) - 1)/2 is a pentagonal number, g(7) = 1, and so a(7) = 1 + 3*a(6) - 5*a(4) + 7*a(1) = 1 + 195 - 100 + 14 = 110. - Peter Bala, Apr 06 2022
a(n) = p(n/2) + Sum_{k \in Z, k != 0} (-1)^{k-1} a(n-k^2), here p(n) = A000041(n) and p(x) = 0 when x is not an integer. - Yen-chi R. Lin, Jun 24 2024
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EXAMPLE
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Assume there are integers of two kinds: k and k'; then a(3) = 10 since 3 has the following partitions into parts of two kinds: 111, 111', 11'1', 1'1'1', 12, 1'2, 12', 1'2', 3, and 3'. - W. Edwin Clark, Jun 24 2011
There are a(4)=20 partitions of 4 into 2 sorts of parts. Here p:s stands for "part p of sort s":
01: [ 1:0 1:0 1:0 1:0 ]
02: [ 1:0 1:0 1:0 1:1 ]
03: [ 1:0 1:0 1:1 1:1 ]
04: [ 1:0 1:1 1:1 1:1 ]
05: [ 1:1 1:1 1:1 1:1 ]
06: [ 2:0 1:0 1:0 ]
07: [ 2:0 1:0 1:1 ]
08: [ 2:0 1:1 1:1 ]
09: [ 2:0 2:0 ]
10: [ 2:0 2:1 ]
11: [ 2:1 1:0 1:0 ]
12: [ 2:1 1:0 1:1 ]
13: [ 2:1 1:1 1:1 ]
14: [ 2:1 2:1 ]
15: [ 3:0 1:0 ]
16: [ 3:0 1:1 ]
17: [ 3:1 1:0 ]
18: [ 3:1 1:1 ]
19: [ 4:0 ]
20: [ 4:1 ]
The a(4)=20 ordered pairs (R,S) of partitions for n=4 are
([4], [])
([3, 1], [])
([2, 2], [])
([2, 1, 1], [])
([1, 1, 1, 1], [])
([3], [1])
([2, 1], [1])
([1, 1, 1], [1])
([2], [2])
([2], [1, 1])
([1, 1], [2])
([1, 1], [1, 1])
([1], [3])
([1], [2, 1])
([1], [1, 1, 1])
([], [4])
([], [3, 1])
([], [2, 2])
([], [2, 1, 1])
([], [1, 1, 1, 1])
This list was created with the Sage command
for P in PartitionTuples(2,4) : print P;
G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 36*x^5 + 65*x^6 + 110*x^7 + 185*x^8 + ...
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MAPLE
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MATHEMATICA
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CoefficientList[ Series[ Product[1/(1 - x^n)^2, {n, 40}], {x, 0, 37}], x]; (* Robert G. Wilson v, Feb 03 2005 *)
Table[Count[Partitions[2*n], q_ /; Tr[(-1)^Mod[Flatten[Position[q, _?OddQ]], 2]] === 0], {n, 12}] (* Wouter Meeussen, Apr 17 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^-2, {x, 0, n}]; (* Michael Somos, Oct 12 2015 *)
Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@n], {n, 0, 15}] (* Robert Price, Jun 15 2020 *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 1 / eta(x + A)^2, n))}; /* Michael Somos, Nov 14 2002 */
(PARI) Vec(1/eta('x+O('x^66))^2) /* Joerg Arndt, Jun 25 2011 */
(Haskell)
a000712 = p a008619_list where
p _ 0 = 1
p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
(Julia) # DedekindEta is defined in A000594.
A000712List(len) = DedekindEta(len, -2)
(SageMath) # uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(0, 1, 2, 2)
b = EulerTransform(a)
(Python)
from sympy import npartitions
def A000712(n): return (sum(npartitions(k)*npartitions(n-k) for k in range(n+1>>1))<<1) + (0 if n&1 else npartitions(n>>1)**2) # Chai Wah Wu, Sep 25 2023
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CROSSREFS
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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EXTENSIONS
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More terms from Joe Keane (jgk(AT)jgk.org), Nov 17 2001
More terms from Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 15 2004
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STATUS
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approved
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