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A087897 Number of partitions of n into odd parts greater than 1. 14
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

C Ballantine, M Merca, Padovan numbers as sums over partitions into odd parts, Journal of Inequalities and Applications, (2016) 2016:1. doi:10.1186/s13660-015-0952-5

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

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

LINKS

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

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.

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.

M. 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(x^(3k)/product(1-x^(2j), j=1..k), k=1..infinity). - 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

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)

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 *)

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

Cf. A000009, A002865.

Sequence in context: A090184 A174575 A029057 * A029056 A226503 A036847

Adjacent sequences:  A087894 A087895 A087896 * A087898 A087899 A087900

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Dec 04 2003

STATUS

approved

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Last modified October 23 03:37 EDT 2018. Contains 316519 sequences. (Running on oeis4.)