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%I #124 Oct 27 2023 15:53:26
%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
%C From _Gus Wiseman_, Aug 22 2021: (Start)
%C 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:
%C 32 222 322 332 432 3322 3332 4332 4432 5432 43332
%C 2222 3222 22222 4322 33222 33322 33332 44322
%C 32222 222222 43222 43322 333222
%C 322222 332222 432222
%C 2222222 3222222
%C (End)
%D J. W. L. Glaisher, Identities, Messenger of Mathematics, 5 (1876), pp. 111-112. see Eq. I
%H Chai Wah Wu, <a href="/A087897/b087897.txt">Table of n, a(n) for n = 0..10000</a> (n = 0..1000 from Alois P. Heinz)
%H C. Ballantine and 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 James Mc Laughlin, Andrew V. Sills, and Peter Zimmer, <a href="https://doi.org/10.37236/36">Rogers-Ramanujan-Slater Type Identities</a>, Electronic J. Combinatorics, DS15, 1-59, May 31, 2008.
%H Michael 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_{k >= 1} x^(3*k)/Product_{j = 1..k} (1 - x^(2*j)). - _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
%F 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
%F 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
%F 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
%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)
%e From _Gus Wiseman_, Feb 16 2021: (Start)
%e 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.
%e 7 53 9 55 B 75 D 77 F 97 H 99 J
%e 333 73 533 93 553 95 555 B5 755 B7 775
%e 3333 733 B3 753 D3 773 D5 955
%e 5333 933 5533 953 F3 973
%e 33333 7333 B33 5553 B53
%e 53333 7533 D33
%e 9333 55333
%e 333333 73333
%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-François 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-François 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 *)
%t Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&OddQ[Times@@#]&]],{n,0,30}] (* _Gus Wiseman_, Feb 16 2021 *)
%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
%o (Python)
%o from functools import lru_cache
%o @lru_cache(maxsize=None)
%o def A087897_T(n,k):
%o if n==0: return 1
%o if k<3 or n<0: return 0
%o return A087897_T(n,k-2)+A087897_T(n-k,k)
%o def A087897(n): return A087897_T(n,n-(n&1^1)) # _Chai Wah Wu_, Sep 23 2023, after _Alois P. Heinz_
%Y The ordered version is A000931.
%Y Partitions with no ones are counted by A002865, ranked by A005408.
%Y The even version is A035363, ranked by A066207.
%Y The version for factorizations is A340101.
%Y Partitions whose only even part is the smallest are counted by A341447.
%Y The Heinz numbers of these partitions are given by A341449.
%Y A000009 counts partitions into odd parts, ranked by A066208.
%Y A025147 counts strict partitions with no 1's.
%Y A025148 counts strict partitions with no 1's or 2's.
%Y A026804 counts partitions whose smallest part is odd, ranked by A340932.
%Y A027187 counts partitions with even length/maximum, ranks A028260/A244990.
%Y A027193 counts partitions with odd length/maximum, ranks A026424/A244991.
%Y A058695 counts partitions of odd numbers, ranked by A300063.
%Y A058696 counts partitions of even numbers, ranked by A300061.
%Y A340385 counts partitions with odd length and maximum, ranked by A340386.
%Y Cf. A000041, A003114, A039900, A160786, A257991/A257992, A264396, A300272, A339662, A339737, A339886.
%K nonn,easy
%O 0,10
%A _N. J. A. Sloane_, Dec 04 2003
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