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A007294 Number of partitions of n into nonzero triangular numbers.
(Formerly M0234)
104
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; text; 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, 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

Number of members of A181818 with a bigomega value of n (cf. A001222). - Matthew Vandermast, May 19 2012

REFERENCES

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

LINKS

T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)

Gert Almkvist, Asymptotics of various partitions, arXiv:math/0612446 [math.NT], 2006.

G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.

N. A. Brigham, A General Asymptotic Formula for Partition Functions, Proc. Amer. Math. Soc., vol. 1 (1950), p. 191.

Zhicheng Gao, Andrew MacFie and Daniel Panario, Counting words by number of occurrences of some patterns, The Electronic Journal of Combinatorics, 18 (2011), #P143.

Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.

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, Journal of Integer Sequences, Vol. 7, 2004.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

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, 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

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.

From Gus Wiseman, May 03 2019: (Start)

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)

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]

(* 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]

(* Clark Kimberling, Mar 09 2014 *)

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-Fran├žois 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 *)

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 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

-- Reinhard Zumkeller, Jun 28 2013

CROSSREFS

Cf. A000217, A051533, A000294.

Cf. A102462.

Row sums of array A176723 and triangle A176724. - Wolfdieter Lang, Jul 19 2010

Cf. A179255 (condition only on differences), A179269 (parts strictly increasing instead of nondecreasing). - Joerg Arndt, Mar 22 2011

Cf. A024940, A280366.

Row sums of A319797.

Cf. A007862, A025487, A240026, A320509, A325324, A325354, A325356, A325362, A325363.

Sequence in context: A029048 A086160 A029047 * A053282 A218084 A240046

Adjacent sequences:  A007291 A007292 A007293 * A007295 A007296 A007297

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Mira Bernstein

EXTENSIONS

Additional comments from Roland Bacher, Jun 17 2001

STATUS

approved

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Last modified April 21 14:00 EDT 2021. Contains 343154 sequences. (Running on oeis4.)