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A007294
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Number of partitions of n into nonzero triangular numbers.
(Formerly M0234)
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115
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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
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OFFSET
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0,4
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COMMENTS
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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, Apr 08 2008
Also partitions with condition as in A179255, and additionally, if more than one 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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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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, Aug 26 2003
For n>0, a(n) is Euler Transform of [1,0,1,0,0,1,0,0,0,1,0,0,0,0,1,...], i.e A010054, n>0. - Benedict W. J. Irwin, Jul 29 2016
a(n) ~ exp(3*Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 2) * Zeta(3/2) / (2^(7/2) * sqrt(3) * Pi * n^(3/2)) [Brigham 1950 (exponential part), Almkvist 2006]. - Vaclav Kotesovec, Dec 31 2016
G.f.: Sum_{i>=0} x^(i*(i+1)/2) / Product_{j=1..i} (1 - x^(j*(j+1)/2)). - Ilya Gutkovskiy, May 07 2017
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EXAMPLE
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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.
The a(1) = 1 through a(9) = 6 partitions using nonzero triangular numbers are the following. The Heinz numbers of these partitions are given by A325363.
1 11 3 31 311 6 61 611 63
111 1111 11111 33 331 3311 333
3111 31111 311111 6111
111111 1111111 11111111 33111
3111111
111111111
The a(1) = 1 through a(10) = 7 partitions with weakly decreasing multiplicities are the following. Equivalent to Matthew Vandermast's comment, the Heinz numbers of these partitions are given by A025487 (products of primorial numbers).
1 11 21 211 2111 321 3211 32111 32211 4321
111 1111 11111 2211 22111 221111 222111 322111
21111 211111 2111111 321111 2221111
111111 1111111 11111111 2211111 3211111
21111111 22111111
111111111 211111111
1111111111
The a(1) = 1 through a(11) = 7 partitions with weakly increasing differences (where the last part is taken to be zero) are the following. The Heinz numbers of these partitions are given by A325362 (A = 10, B = 11).
(1) (2) (3) (4) (5) (6) (7) (8) (9) (A) (B)
(21) (31) (41) (42) (52) (62) (63) (73) (83)
(51) (61) (71) (72) (82) (92)
(321) (421) (521) (81) (91) (A1)
(531) (631) (731)
(621) (721) (821)
(4321) (5321)
(End)
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MAPLE
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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))):
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:
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MATHEMATICA
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CoefficientList[ Series[ 1/Product[1 - x^(i(i + 1)/2), {i, 1, 50}], {x, 0, 70}], x]
(* also *)
t = Table[n (n + 1)/2, {n, 1, 200}] ; p[n_] := IntegerPartitions[n, All, t]; Table[p[n], {n, 0, 12}] (*shows partitions*)
a[n_] := Length@p@n; a /@Range[0, 80]
b[n_, i_] := b[n, i] = Which[n < 0, 0, n == 0, 1, i == 0, 0, True, b[n, i-1]+b[n-i*(i+1)/2, i]]; a[n_] := b[n, Floor[Sqrt[2*n]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 09 2014, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n], OrderedQ[Differences[Append[#, 0]]]&]], {n, 0, 30}] (* Gus Wiseman, May 03 2019 *)
nmax = 58; t = Table[PolygonalNumber[n], {n, nmax}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[t, x]], {n, 0, nmax}] (* Robert Price, Aug 02 2020 *)
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PROG
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(Sage)
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 len([1 for x in Partitions(n, max_slope=-1) if allowed(x[::-1])]) # D. S. McNeil, Jan 06 2011
(PARI) N=66; Vec(1/prod(k=1, N, 1-x^(k*(k+1)\2))+O(x^N)) \\ Joerg Arndt, Apr 14 2013
(Haskell)
a007294 = p $ tail a000217_list where
p _ 0 = 1
p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
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CROSSREFS
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Cf. A179255 (condition only on differences), A179269 (parts strictly increasing instead of nondecreasing). - Joerg Arndt, Mar 22 2011
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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