0,4

Also number of decreasing integer sequences l(1) >= l(2) >= l(3) >= .. 0 such that sum('i*l(i)','i'=1..infinity)=n.

a(n) is also the number of partitions of n such that #{parts equal to i} >= #{parts equal to j} if i <= j.

Also the number of partitions of n (necessarily into distinct parts) where the part sizes are monotonically decreasing (including the last part, which is the difference between the last part and a "part" of size 0). These partitions are the conjugates of the partitions with number of parts of size i increasing. - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Apr 08 2008

G. E. Andrews, MacMahon's partition analysis: II, Fundamental theorems, Annals of Combinatorics, 4 (2000), 327-338.

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

T. D. Noe, Table of n, a(n) for n = 0..1000

James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.

Jan Snellman and Michael Paulsen, Enumeration of Concave Integer Partitions, J. Integer Seqs., Vol. 7, 2004.

Product('(1-z^binomial(k, 2))^(-1)', 'k'=2..infinity);

G.f.: Product ( 1 - x^(k(k-1)/2))^(-1). - Les Reid, Jul 26, 2002

For n>0: a(n) = b(n, 1) where b(n, k) = if n>k*(k+1)/2 then b(n-k*(k+1)/2, k) + b(n, k+1) else (if n=k*(k+1)/2 then 1 else 0). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 26 2003

6=3+3=3+1+1+1=1+1+1+1+1+1 so a(6) = 4.

a(7)=4: Four sequences as above are (7,0,..), (5,1,0,..), (3,2,0,..),(2,1,1,0,..). They correspond to the partitions 1^7, 2 1^5, 2^2 1^3, 3 2 1^2 of seven or in the main description to the partitions 1^7, 3 1^4, 3^2 1, 6 1.

CoefficientList[ Series[ 1/Product[1 - x^(i(i + 1)/2), {i, 1, 50}], {x, 0, 70}], x]

Cf. A000217, A051533, A000294.

Cf. A102462.

Sequence in context: A029048 A086160 A029047 this_sequence A053282 A001584 A112801

Adjacent sequences: A007291 A007292 A007293 this_sequence A007295 A007296 A007297

nonn

N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein (mira(AT)math.berkeley.edu)

Additional comments from Roland Bacher (Roland.Bacher(AT)ujf-grenoble.fr), Jun 17 2001