%I A000219 M2566 N1016
%S A000219 1,1,3,6,13,24,48,86,160,282,500,859,1479,2485,4167,6879,11297,
%T A000219 18334,29601,47330,75278,118794,186475,290783,451194,696033,1068745,
%U A000219 1632658,2483234,3759612,5668963,8512309,12733429,18974973,28175955
%N A000219 Number of planar partitions of n.
%C A000219 Two-dimensional partitions of n in which no row or column is longer than 
               the one before it (compare A001970). E.g. a(4) = 13:
%C A000219 4.31.3.22.2.211.21..2.1111.111.11.11.1 but not 2
%C A000219 .....1....2.....1...1......1...11.1..1........ 11
%C A000219 ....................1.............1..1
%C A000219 .....................................1
%C A000219 Can also be regarded as number of "safe pilings" of cubes in the corner 
               of a room: the height should not increase away from the corner - 
               Wouter Meeussen (wouter.meeussen(AT)pandora.be).
%C A000219 Also number of partitions of n objects of 2 colors, each part containing 
               at least one black object. - (Christian G. Bower (bowerc(AT)usa.net), 
               Jan 08 2004)
%C A000219 Number of partitions of n into 1 type of part 1, 2 types of part 2, ..., 
               k types of part k. e.g. n=3 gives 111, 12, 12', 3, 3', 3''. - Jon 
               Perry (perry(AT)globalnet.co.uk), May 27 2004
%C A000219 Can also be regarded as the number of Jordan canonical forms for an nxn 
               matrix. (i.e. a 5x5 matrix has 24 distinct Jordan canonical forms, 
               dependent on the algebraic and geometric multiplicity of each eigenvalue.) 
               [From Aaron Gable (agable(AT)hmc.edu), May 26 2009]
%C A000219 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 13 2009: 
               (Start)
%C A000219 1/n * convolution product of n terms * A001157 (sum of squares of divisors 
               of n): (1, 5, 10, 21, 26, 50, 50, 85,...) = a(n). As shown by [Bressoud, 
               p.12]: 1/6 * [1*24 + 5*13 + 10*6 + 21*3 + 26*1 + 50*1] = 288/6 = 
               48.
%C A000219 Convolved with the aerated version (1, 0, 1, 0, 3, 0, 6, 0, 13,...) = 
               A026007: (1, 1, 2, 5, 8, 16, 28, 49, 83,...). (End)
%C A000219 Starting with offset 1 = row sums of triangle A162453 [From Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Jul 03 2009]
%C A000219 Also number of functions from an n-set to itself, up to permutation of 
               the set (compare to partition function, which is number of conjugacy 
               classes in S_n) [From Harry Altman (haltman(AT)umich.edu), Nov 21 
               2009]
%D A000219 G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. 
               Math., 7 (No. 4, 1998), pp. 343-359.
%D A000219 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 241.
%D A000219 A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations 
               for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 
               1097-1100.
%D A000219 Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: 
               MIT Artificial Intelligence Laboratory, Memo AIM-239, Item 18, Feb. 
               1972.
%D A000219 Bender, E. A. and Knuth, D. E. ``Enumeration of Plane Partitions.'' J. 
               Combin. Theory A. 13, 40-54, 1972.
%D A000219 D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; pp(n) 
               on p. 10.
%D A000219 D. M. Bressoud and J. Propp, How the alternating sign matrix conjecture 
               was solved, Notices Amer. Math. Soc., 46 (No. 6, 1999), 637-646.
%D A000219 L. Carlitz, Generating functions and partition problems, pp. 144-169 
               of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 
               8 (1965). Amer. Math. Soc., see p. 145, eq. (1.6).
%D A000219 D. E. Knuth, A Note on Solid Partitions. Math. Comp. 24, 955-961, 1970.
%D A000219 P. A. MacMahon, Memoir on the theory of partitions of numbers - Part 
               VI, Phil. Trans. Roal Soc., 211 (1912), 345-373.
%D A000219 P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and 
               New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
%D A000219 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000219 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000219 E. M. Wright, Rotatable partitions, J. London Math. Soc., 43 (1968), 
               501-505.
%H A000219 T. D. Noe, <a href="b000219.txt">Table of n, a(n) for n = 0..400</a>
%H A000219 G. Almkvist, <a href="http://www.expmath.org/expmath/volumes/7/7.html">
               Asymptotic formulas and generalized Dedekind sums</a>, Exper. Math., 
               7 (No. 4, 1998), pp. 343-359.
%H A000219 G. E. Andrews, P. Paule, <a href="http://dx.doi.org/10.1112/jlms/jdm079">
               MacMahon's partition analysis XII: Plane Partitions</a>, J. Lond. 
               Math. Soc., 76 (2007), 647-666. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Jan 19 2009]
%H A000219 Beeler, M., Gosper, R. W. and Schroeppel, R., <a href="http://www.inwap.com/
               pdp10/hbaker/hakmem/boolean.html#item18">HAKMEM, ITEM 18</a>
%H A000219 H. Bottomley, <a href="a219.gif">Illustration of initial terms</a>
%H A000219 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
               Publications/books.html">Analytic Combinatorics</a>, 2009; see page 
               580
%H A000219 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=141">
               Encyclopedia of Combinatorial Structures 141</a>
%H A000219 P. A. MacMahon, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;
               idno=ABU9009">Combinatory analysis</a>.
%H A000219 Ville Mustonen and R. Rajesh, <a href="http://arXiv.org/abs/cond-mat/
               0303607">Numerical Estimation of the Asymptotic Behaviour of Solid 
               Partitions ...</a>
%H A000219 L. Mutafchiev and E. Kamenov, <a href="http://arXiv.org/abs/math.CO/0601253">
               On The Asymptotic Formula for the Number of Plane Partitions...</
               a>
%H A000219 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%H A000219 J. Stienstra, <a href="http://arXiv.org/abs/math.NT/0502197">Mahler measure, 
               Eisenstein series and dimers</a>
%H A000219 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PlanePartition.html">Link to a section of The World of Mathematics.</
               a>
%H A000219 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000219 G.f.: Product_{k >= 1} 1/(1 - x^k)^k. - MacMahon, 1912.
%F A000219 Euler transform of sequence [1, 2, 3, ...].
%F A000219 a(n) ~ (c_2 / n^(25/26)) * exp( c_1 * n^(2/3) ), where c_1 = 2.00945... 
               and c_2 = 0.40099... - Wright, 1931.
%F A000219 a(n)=1/n*Sum_{k=1..n} a(n-k)*sigma_2(k), n > 0, a(0)=1, where sigma_2(n)=A001157(n)=sum 
               of squares of divisors of n. - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Jan 20 2002
%F A000219 G.f.: exp(Sum_{n>0} sigma_2(n)*x^n/n). a(n) = Sum_{pi} Product_{i=1..n} 
               binomial(k(i)+i-1, k(i)) where pi runs through all nonnegative solutions 
               of k(1)+2*k(2)+..+n*k(n)=n. - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Jan 10 2003
%e A000219 A planar partition of 13:
%e A000219 4 3 1 1
%e A000219 2 1
%e A000219 1
%e A000219 a(5) = (1/5!)*(sigma_2(1)^5+10*sigma_2(2)*sigma_2(1)^3+20*sigma_2(3)*sigma_2(1)^2+15*sigma_2(1)*sigma_2(2)^2+\
               30*sigma_2(4)*sigma_2(1)+20*sigma_2(2)*sigma_2(3)+24*sigma_2(5)) 
               = 24. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 10 2003
%p A000219 series(mul((1-x^k)^(-k),k=1..64),x,63);
%t A000219 Rest@CoefficientList[ Series[ Product[ (1-x^k)^-k, {k, 1, 64} ], {x, 
               0, 64} ], x ]
%o A000219 (PARI) a(n)=if(n<0,0,polcoeff(exp(sum(k=1,n,x^k/(1-x^k)^2/k,x*O(x^n))),
               n)) /* Michael Somos Jan 29 2005 */
%o A000219 (PARI) a(n)=if(n<0,0,polcoeff(prod(k=1,n,(1-x^k+x*O(x^n))^-k),n)) /* 
               Michael Somos Jan 29 2005 */
%o A000219 Contribution from R. J. Mathar (mathar(AT)st.rw.leidenuniv.nl), Oct 18 
               2009: (Start)
%o A000219 (Python) def divisors(n):
%o A000219 ...a = {1,n}
%o A000219 ...for x in range(2,n//2+1):
%o A000219 ......if n % x == 0:
%o A000219 .........a |= {x}
%o A000219 ...return a
%o A000219 def sigma(n,k):
%o A000219 ...a=0
%o A000219 ...for d in divisors(n):
%o A000219 ......a += d**k
%o A000219 ...return a
%o A000219 def A000219(n):
%o A000219 ...if n <=1:
%o A000219 ......return 1
%o A000219 ...else:
%o A000219 ......a=0
%o A000219 ......for k in range (1,n+1):
%o A000219 .........a += A000219(n-k)*sigma(k,2)
%o A000219 ......return a//n
%o A000219 print([A000219(n) for n in range(0,20)]) (End)
%Y A000219 Cf. A000784, A000785, A000786, A005380, A005987, A048141, A048142, A089300.
%Y A000219 Cf. A023871-A023878.
%Y A000219 Row sums of A089353 and A091438.
%Y A000219 Cf. A026007, A001157.
%Y A000219 A162453 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 03 2009]
%Y A000219 Sequence in context: A018081 A001452 A005405 this_sequence A027999 A005196 
               A032287
%Y A000219 Adjacent sequences: A000216 A000217 A000218 this_sequence A000220 A000221 
               A000222
%K A000219 nonn,nice,easy,core,new
%O A000219 0,3
%A A000219 N. J. A. Sloane (njas(AT)research.att.com).
%E A000219 Corrected Jul 29 2006