|
| |
|
|
A000712
|
|
Number of partitions of n into parts of 2 kinds.
(Formerly M1376 N0536)
|
|
87
| |
|
|
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
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| 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 (vladeta(AT)eunet.rs), Apr 21 2005
Convolution of partition numbers (A000041) with itself. - Graeme McRae (g_m(AT)mcraefamily.com), Jun 07 2006
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 (FrankTAW(AT)Netscape.net), 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,...]. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 26 2008]
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 01 2010: (Start)
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 + ...). (End)
Equals row sums of triangle A175012 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 03 2010]
|
|
|
REFERENCES
| W. Y. C. Chen, K. Q. Ji and H. S. Wilf, BG-ranks and 2-cores, arXiv:math.CO/0605474.
W. Edwin Clark, Mohamed Elhamdadi, Xiang-dong Hou, Masahico Saito and Timothy Yeatman, Connected Quandles Associated with Pointed Abelian Groups, Arxiv preprint arXiv:1107.5777, 2011
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.
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=0..500
E. R. Canfield, C. D. Savage and H. S. Wilf, Regularly spaced subsums of integer partitions
W. Edwin Clark and Xiang-dong Hou, Galkin Quandles, Pointed Abelian Groups, and Sequence A000712 (arXiv:1108.2215v1 [math.CO]), Aug 10, 2011 [added by Jonathan Vos Post]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 129
N. J. A. Sloane, Transforms
Index entries for expansions of Product_{k >= 1} (1-x^k)^m
|
|
|
FORMULA
| G.f.: 1/prod(m>=1, 1-x^m )^2.
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(n) = A006330(n) + A001523(n). - 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 5 2011
a(n) ~ 1/12*3^(1/4)*n^(-5/4)*exp(2/3*3^(1/2)*pi*n^(1/2)) - Joe Keane (jgk(AT)jgk.org), Sep 13 2002
G.f. : product(i=1, oo, (1+x^i)^(2*A001511(i))) (see A000041) - Jon Perry (perry(AT)globalnet.co.uk), Jun 06 2004
|
|
|
EXAMPLE
| 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]
|
|
|
MAPLE
| with(combinat): A000712:=n->sum(numbpart(k)*numbpart(n-k), k=0..n): seq(A000712(n), n=0..37); (E. Deutsch)
|
|
|
MATHEMATICA
| CoefficientList[ Series[ Product[1/(1 - x^n)^2, {n, 40}], {x, 0, 37}], x] (from Robert G. Wilson v Feb 03 2005)
|
|
|
PROG
| (PARI) {a(n) = local(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 */
|
|
|
CROSSREFS
| Cf. A000165, A000041, A002107 (reciprocal of g.f.).
Cf. A002720.
A000716, A010815 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 26 2008]
Cf. A175012 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 03 2010]
Sequence in context: A103928 A103929 A121597 * A032442 A102688 A001629
Adjacent sequences: A000709 A000710 A000711 * A000713 A000714 A000715
|
|
|
KEYWORD
| nonn,easy,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| 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
|
| |
|
|
