A087897 - OEIS

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

From Gus Wiseman, Aug 22 2021: (Start)

Also the number of integer partitions of n - 1 whose parts cover an interval of positive integers starting with 2. These partitions are ranked by A339886. For example, the a(6) = 1 through a(16) = 5 partitions are:

32 222 322 332 432 3322 3332 4332 4432 5432 43332

2222 3222 22222 4322 33222 33322 33332 44322

32222 222222 43222 43322 333222

322222 332222 432222

2222222 3222222

(End)

REFERENCES

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

LINKS

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (n = 0..1000 from Alois P. Heinz)

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 Mc Laughlin et al., Section 1.3, Entry 7.) - Peter Bala, Feb 02 2021

a(2*n+1) = Sum{j>=1} A008284(n+1-j,2*j - 1) and a(2*n) = Sum{j>=1} A008284(n-j, 2*j). - Gregory L. Simay, Sep 22 2023

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(19) = 8 partitions are the following (A..J = 10..19). The Heinz numbers of these partitions are given by A341449.

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

333 73 533 93 553 95 555 B5 755 B7 775

3333 733 B3 753 D3 773 D5 955

5333 933 5533 953 F3 973

33333 7333 B33 5553 B53

53333 7533 D33

9333 55333

333333 73333

(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

(Python)

from functools import lru_cache

@lru_cache(maxsize=None)

def A087897_T(n, k):

if n==0: return 1

if k<3 or n<0: return 0

return A087897_T(n, k-2)+A087897_T(n-k, k)

def A087897(n): return A087897_T(n, n-(n&1^1)) # Chai Wah Wu, Sep 23 2023, after Alois P. Heinz

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, ranks A028260/A244990.

A027193 counts partitions with odd length/maximum, ranks 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, A264396, A300272, A339662, A339737, A339886.

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