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A087897 Number of partitions of n into odd parts greater than 1. 28
1, 0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 8, 8, 10, 12, 13, 15, 18, 20, 23, 27, 30, 34, 40, 44, 50, 58, 64, 73, 83, 92, 104, 118, 131, 147, 166, 184, 206, 232, 256, 286, 320, 354, 394, 439, 485, 538, 598, 660, 730, 809, 891, 984, 1088, 1196, 1318, 1454, 1596, 1756 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,10

COMMENTS

Also number of partitions of n into distinct parts which are not powers of 2.

Also number of partitions of n into distinct parts such that the two largest parts differ by 1.

Also number of partitions of n such that the largest part occurs an odd number of times that is at least 3 and every other part occurs an even number of times. Example: a(10) = 2 because we have [2,2,2,1,1,1,1] and [2,2,2,2,2]. - Emeric Deutsch, Mar 30 2006

Also difference between number of partitions of 1+n into distinct parts and number of partitions of n into distinct parts. - Philippe LALLOUET, May 08 2007

In the Berndt reference replace {a -> -x, q -> x} in equation (3.1) to get f(x). G.f. is 1 - x * (1 - f(x)).

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Also number of symmetric unimodal compositions of n+3 where the maximal part appears three times. - Joerg Arndt, Jun 11 2013

Let c(n) = number of palindromic partitions of n whose greatest part has multiplicity 3; then c(n) = a(n-3) for n>=3. - Clark Kimberling, Mar 05 2014

REFERENCES

J. W. L. Glaisher, Identities, Messenger of Mathematics, 5 (1876), pp. 111-112. see Eq. I

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

C. Ballantine and M. Merca, Padovan numbers as sums over partitions into odd parts, Journal of Inequalities and Applications, (2016) 2016:1; doi.

B. C. Berndt, B. Kim, and A. J. Yee, Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions, J. Comb. Thy. Ser. A, 117 (2010), 957-973.

Howard D. Grossman, Problem 228, Mathematics Magazine, 28 (1955), p. 160.

R. K. Guy, Two theorems on partitions, Math. Gaz., 42 (1958), 84-86. Math. Rev. 20 #3110.

Cristiano Husu, The butterfly sequence: the second difference sequence of the numbers of integer partitions with distinct parts, its pentagonal number structure, its combinatorial identities and the cyclotomic polynomials 1-x and 1+x+x^2, arXiv:1804.09883 [math.NT], 2018.

James Mc Laughlin, Andrew V. Sills, and Peter Zimmer, Rogers-Ramanujan-Slater Type Identities, Electronic J. Combinatorics, DS15, 1-59, May 31, 2008.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q^(-1/24) * (1 - q) * eta(q^2) / eta(q) in powers of q.

Expansion of (1 - x) / chi(-x) in powers of x where chi() is a Ramanujan theta function.

G.f.: 1 + x^3 + x^5*(1 + x) + x^7*(1 + x)*(1 + x^2) + x^9*(1 + x)*(1 + x^2)*(1 + x^3) + ... [Glaisher 1876]. - Michael Somos, Jun 20 2012

G.f.: Product_{k >= 1} 1/(1-x^(2*k+1)).

G.f.: Product_{k >= 1, k not a power of 2} (1+x^k).

G.f.: Sum_{k >= 1} x^(3*k)/Product_{j = 1..k} (1 - x^(2*j)). - Emeric Deutsch, Mar 30 2006

a(n) ~ exp(Pi*sqrt(n/3)) * Pi / (8 * 3^(3/4) * n^(5/4)) * (1 - (15*sqrt(3)/(8*Pi) + 11*Pi/(48*sqrt(3)))/sqrt(n) + (169*Pi^2/13824 + 385/384 + 315/(128*Pi^2))/n). - Vaclav Kotesovec, Aug 30 2015, extended Nov 04 2016

G.f.: 1/(1 - x^3) * Sum_{n >= 0} x^(5*n)/Product_{k = 1..n} (1 - x^(2*k)) = 1/((1 - x^3)*(1 - x^5)) * Sum_{n >= 0} x^(7*n)/Product_{k = 1..n} (1 - x^(2*k)) = ..., extending Deutsch's result dated Mar 30 2006. - Peter Bala, Jan 15 2021

G.f.: Sum_{n >= 0} x^(n*(2*n+1))/Product_{k = 2..2*n+1} (1 - x^k). (Set z = x^3 and q = x^2 in McLaughlin et al., Section 1.3, Entry 7.) - Peter Bala, Feb 02 2021

EXAMPLE

1 + x^3 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + 2*x^10 + 2*x^11 + 3*x^12 + 3*x^13 + ...

q + q^73 + q^121 + q^145 + q^169 + q^193 + 2*q^217 + 2*q^241 + 2*q^265 + ...

a(10)=2 because we have [7,3] and [5,5].

From Joerg Arndt, Jun 11 2013: (Start)

There are a(22)=13 symmetric unimodal compositions of 22+3=25 where the maximal part appears three times:

01:  [ 1 1 1 1 1 1 1 1 3 3 3 1 1 1 1 1 1 1 1 ]

02:  [ 1 1 1 1 1 1 2 3 3 3 2 1 1 1 1 1 1 ]

03:  [ 1 1 1 1 1 5 5 5 1 1 1 1 1 ]

04:  [ 1 1 1 1 2 2 3 3 3 2 2 1 1 1 1 ]

05:  [ 1 1 1 2 5 5 5 2 1 1 1 ]

06:  [ 1 1 2 2 2 3 3 3 2 2 2 1 1 ]

07:  [ 1 1 3 5 5 5 3 1 1 ]

08:  [ 1 1 7 7 7 1 1 ]

09:  [ 1 2 2 5 5 5 2 2 1 ]

10:  [ 1 4 5 5 5 4 1 ]

11:  [ 2 2 2 2 3 3 3 2 2 2 2 ]

12:  [ 2 3 5 5 5 3 2 ]

13:  [ 2 7 7 7 2 ]

(End)

From Gus Wiseman, Feb 16 2021: (Start)

The a(7) = 1 through a(20) = 10 partitions are the following (A..H = 10..18). The Heinz numbers of these partitions are given by A341449.

  7  53  9    55  B    75    D    77    F      97    H      99      J      B9

         333  73  533  93    553  95    555    B5    755    B7      775    D7

                       3333  733  B3    753    D3    773    D5      955    F5

                                  5333  933    5533  953    F3      973    H3

                                        33333  7333  B33    5553    B53    5555

                                                     53333  7533    D33    7553

                                                            9333    55333  7733

                                                            333333  73333  9533

                                                                           B333

                                                                           533333

(End)

MAPLE

To get 128 terms: t4 := mul((1+x^(2^n)), n=0..7); t5 := mul((1+x^k), k=1..128): t6 := series(t5/t4, x, 100); t7 := seriestolist(t6);

# second Maple program:

b:= proc(n, i) option remember; `if`(n=0, 1,

      `if`(i<3, 0, b(n, i-2)+`if`(i>n, 0, b(n-i, i))))

    end:

a:= n-> b(n, n-1+irem(n, 2)):

seq(a(n), n=0..80);  # Alois P. Heinz, Jun 11 2013

MATHEMATICA

max = 65; f[x_] := Product[ 1/(1 - x^(2k+1)), {k, 1, max}]; CoefficientList[ Series[f[x], {x, 0, max}], x] (* Jean-François Alcover, Dec 16 2011, after Emeric Deutsch *)

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<3, 0, b[n, i-2]+If[i>n, 0, b[n-i, i]]] ]; a[n_] := b[n, n-1+Mod[n, 2]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Apr 01 2015, after Alois P. Heinz *)

Flatten[{1, Table[PartitionsQ[n+1] - PartitionsQ[n], {n, 0, 80}]}] (* Vaclav Kotesovec, Dec 01 2015 *)

Table[Length[Select[IntegerPartitions[n], FreeQ[#, 1]&&OddQ[Times@@#]&]], {n, 0, 30}] (* Gus Wiseman, Feb 16 2021 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - x) * eta(x^2 + A) / eta(x + A), n))} /* Michael Somos, Nov 13 2011 */

(Haskell)

a087897 = p [3, 5..] where

   p [] _ = 0

   p _  0 = 1

   p ks'@(k:ks) m | m < k     = 0

                  | otherwise = p ks' (m - k) + p ks m

-- Reinhard Zumkeller, Aug 12 2011

CROSSREFS

The ordered version is A000931.

Partitions with no ones are counted by A002865, ranked by A005408.

The even version is A035363, ranked by A066207.

The version for factorizations is A340101.

Partitions whose only even part is the smallest are counted by A341447.

The Heinz numbers of these partitions are given by A341449.

A000009 counts partitions into odd parts, ranked by A066208.

A025147 counts strict partitions with no 1's.

A025148 counts strict partitions with no 1's or 2's.

A026804 counts partitions whose smallest part is odd, ranked by A340932.

A027187 counts partitions with even length/maximum, ranked by A028260/A244990.

A027193 counts partitions with odd length/maximum, ranked by A026424/A244991.

A058695 counts partitions of odd numbers, ranked by A300063.

A058696 counts partitions of even numbers, ranked by A300061.

A340385 counts partitions with odd length and maximum, ranked by A340386.

Cf. A000041, A003114, A039900, A160786, A257991/A257992, A300272.

Sequence in context: A090184 A174575 A029057 * A029056 A226503 A036847

Adjacent sequences:  A087894 A087895 A087896 * A087898 A087899 A087900

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Dec 04 2003

STATUS

approved

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Last modified May 16 01:55 EDT 2021. Contains 343937 sequences. (Running on oeis4.)