|
| |
|
|
A007294
|
|
Number of partitions of n into nonzero triangular numbers.
(Formerly M0234)
|
|
36
| |
|
|
1, 1, 1, 2, 2, 2, 4, 4, 4, 6, 7, 7, 10, 11, 11, 15, 17, 17, 22, 24, 25, 32, 35, 36, 44, 48, 50, 60, 66, 68, 81, 89, 92, 107, 117, 121, 141, 153, 159, 181, 197, 205, 233, 252, 262, 295, 320, 332, 372, 401, 417, 465, 501, 520, 575, 619, 645, 710, 763
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,4
|
|
|
COMMENTS
| 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
Also partitions with condition as in A179255, and additionally, if more than 1 part, first difference >= first part: for example, a(10)=7 as there are 7 such partitions of 10: 1+2+3+4= 1+2+7= 1+3+6= 1+9= 2+8= 3+7= 10. [Joerg Arndt, Mar 22 2011].
|
|
|
REFERENCES
| G. E. Andrews, MacMahon's partition analysis: II, Fundamental theorems, Annals of Combinatorics, 4 (2000), 327-338.
Zhicheng Gao, Andrew MacFie and Daniel Panario, Counting words by number of occurrences of some patterns, The Electronic Journal of Combinatorics, 18 (2011), #p143; http://www.combinatorics.org/Volume_18/PDF/v18i1p143.pdf.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
| 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.
|
|
|
FORMULA
| G.f.: 1/Product(k>=2, 1-z^binomial(k, 2)).
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
|
|
|
EXAMPLE
| 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.
|
|
|
MAPLE
| b:= proc(n, i) option remember;
if n<0 then 0
elif n=0 then 1
elif i=0 then 0
else b(n, i-1) +b(n-i*(i+1)/2, i)
fi
end:
a:= n-> b(n, floor (sqrt (2*n))):
seq (a(n), n=0..100); # Alois P. Heinz, Mar 22 2011
isNondecrP :=proc(L) slp := DIFF(DIFF(L)) ; min(op(%)) >= 0 ; end proc:
A007294 := proc(n) local a, p; a := 0 ; if n = 0 then return 1 ; end if; for p in combinat[partition](n) do if nops(p) = nops(convert(p, set)) then if isNondecrP(p) then if nops(p) =1 then a := a+1 ; elif op(2, p) >= 2*op(1, p) then a := a+1; end if; end if; end if; end do; a ; end proc:
seq(A007294(n), n=0..30) ; # R. J. Mathar, Jan 07 2011
|
|
|
MATHEMATICA
| CoefficientList[ Series[ 1/Product[1 - x^(i(i + 1)/2), {i, 1, 50}], {x, 0, 70}], x]
|
|
|
PROG
| (Sage)
def A007294(n):
....has_nondecreasing_diffs = lambda x: min(differences(x, 2)) >= 0
....special = lambda x: (x[1]-x[0]) >= x[0]
....allowed = lambda x: (len(x) < 2 or special(x)) and (len(x) < 3 or has_nondecreasing_diffs(x))
....return Partitions(n, max_slope=-1).filter(lambda x: allowed(x[::-1])).cardinality() # [D. S. McNeil, Jan 06 2011]
|
|
|
CROSSREFS
| Cf. A000217, A051533, A000294.
Cf. A102462.
Row sums of array A176723 and triangle A176724. [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jul 19 2010]
Cf. A179255 (condition only on differences), A179269 (parts strictly increasing instead of non-decreasing) [Joerg Arndt Mar 22 2011].
Sequence in context: A029048 A086160 A029047 * A053282 A001584 A180019
Adjacent sequences: A007291 A007292 A007293 * A007295 A007296 A007297
|
|
|
KEYWORD
| nonn,changed
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein (mira(AT)math.berkeley.edu)
|
|
|
EXTENSIONS
| Additional comments from Roland Bacher (Roland.Bacher(AT)ujf-grenoble.fr), Jun 17 2001
|
| |
|
|
