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A087897 Number of partitions of n into odd parts greater than 1. 22

%I

%S 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,

%T 34,40,44,50,58,64,73,83,92,104,118,131,147,166,184,206,232,256,286,

%U 320,354,394,439,485,538,598,660,730,809,891,984,1088,1196,1318,1454,1596,1756

%N Number of partitions of n into odd parts greater than 1.

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

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

%C 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

%C 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

%C 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)).

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

%C Also number of symmetric unimodal compositions of n+3 where the maximal part appears three times. - _Joerg Arndt_, Jun 11 2013

%C 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

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

%H Alois P. Heinz, <a href="/A087897/b087897.txt">Table of n, a(n) for n = 0..1000</a>

%H C. Ballantine, M. Merca, <a href="https://www.researchgate.net/publication/289250007_Padovan_numbers_as_sums_over_partitions_into_odd_parts">Padovan numbers as sums over partitions into odd parts</a>, Journal of Inequalities and Applications, (2016) 2016:1; <a href="https://doi.org/10.1186/s13660-015-0952-5">doi</a>.

%H B. C. Berndt, B. Kim, and A. J. Yee, <a href="http://dx.doi.org/10.1016/j.jcta.2009.07.005">Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions</a>, J. Comb. Thy. Ser. A, 117 (2010), 957-973.

%H Howard D. Grossman, <a href="https://www.jstor.org/stable/3029861">Problem 228</a>, Mathematics Magazine, 28 (1955), p. 160.

%H R. K. Guy, <a href="http://www.jstor.org/stable/3609388">Two theorems on partitions</a>, Math. Gaz., 42 (1958), 84-86. Math. Rev. 20 #3110.

%H Cristiano Husu, <a href="https://arxiv.org/abs/1804.09883">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</a>, arXiv:1804.09883 [math.NT], 2018.

%H M. Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

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

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

%F 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

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

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

%F G.f.: sum(x^(3k)/product(1-x^(2j), j=1..k), k=1..infinity). - _Emeric Deutsch_, Mar 30 2006

%F 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

%e 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 + ...

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

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

%e From _Joerg Arndt_, Jun 11 2013: (Start)

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

%e 01: [ 1 1 1 1 1 1 1 1 3 3 3 1 1 1 1 1 1 1 1 ]

%e 02: [ 1 1 1 1 1 1 2 3 3 3 2 1 1 1 1 1 1 ]

%e 03: [ 1 1 1 1 1 5 5 5 1 1 1 1 1 ]

%e 04: [ 1 1 1 1 2 2 3 3 3 2 2 1 1 1 1 ]

%e 05: [ 1 1 1 2 5 5 5 2 1 1 1 ]

%e 06: [ 1 1 2 2 2 3 3 3 2 2 2 1 1 ]

%e 07: [ 1 1 3 5 5 5 3 1 1 ]

%e 08: [ 1 1 7 7 7 1 1 ]

%e 09: [ 1 2 2 5 5 5 2 2 1 ]

%e 10: [ 1 4 5 5 5 4 1 ]

%e 11: [ 2 2 2 2 3 3 3 2 2 2 2 ]

%e 12: [ 2 3 5 5 5 3 2 ]

%e 13: [ 2 7 7 7 2 ]

%e (End)

%p 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);

%p # second Maple program:

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

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

%p end:

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

%p seq(a(n), n=0..80); # _Alois P. Heinz_, Jun 11 2013

%t max = 65; f[x_] := Product[ 1/(1 - x^(2k+1)), {k, 1, max}]; CoefficientList[ Series[f[x], {x, 0, max}], x] (* _Jean-Fran├žois Alcover_, Dec 16 2011, after _Emeric Deutsch_ *)

%t 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-Fran├žois Alcover_, Apr 01 2015, after _Alois P. Heinz_ *)

%t Flatten[{1, Table[PartitionsQ[n+1] - PartitionsQ[n], {n, 0, 80}]}] (* _Vaclav Kotesovec_, Dec 01 2015 *)

%o (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 */

%o (Haskell)

%o a087897 = p [3,5..] where

%o p [] _ = 0

%o p _ 0 = 1

%o p ks'@(k:ks) m | m < k = 0

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

%o -- _Reinhard Zumkeller_, Aug 12 2011

%Y Cf. A000009, A002865.

%K nonn

%O 0,10

%A _N. J. A. Sloane_, Dec 04 2003

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Last modified January 21 11:07 EST 2020. Contains 331105 sequences. (Running on oeis4.)