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A097566 Number of partitions p of n for which Odd(p) = Odd(p') (mod 4), where p' is the conjugate of p.

%I A097566 #45 Mar 12 2021 22:24:43
%S A097566 1,1,0,1,5,5,1,5,20,20,6,20,65,65,25,66,185,185,85,190,481,482,250,
%T A097566 501,1165,1170,666,1230,2666,2685,1646,2850,5827,5887,3830,6303,12251,
%U A097566 12415,8487,13395,24912,25323,18052,27507,49215,50176,37072,54832,94781,96905
%N A097566 Number of partitions p of n for which Odd(p) = Odd(p') (mod 4), where p' is the conjugate of p.
%C A097566 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A097566 Odd(p) is the number of odd parts of a partition p. a(n) is denoted t(n) in Problem 10969.
%H A097566 G. C. Greubel, <a href="/A097566/b097566.txt">Table of n, a(n) for n = 0..1000</a>
%H A097566 George E. Andrews, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v11i2r1">On a Partition Function of Richard Stanley</a>, The Electronic Journal of Combinatorics, Volume 11, Issue 2 (2004-6) (The Stanley Festschrift volume), Research Paper #R1.
%H A097566 M. Ishikawa and J. Zeng, <a href="https://doi.org/10.1016/j.disc.2007.12.064">The Andrews-Stanley partition function and Al-Salam-Chihara polynomials</a>, Disc. Math., 309 (2009), 151-175. (See t(n) p. 151. Note that there is a typo in the g.f. for f(n) - see A144558.) [Added by _N. J. A. Sloane_, Jan 25 2009.]
%H A097566 Andrew V. Sills, <a href="https://doi.org/10.1155/S0161171204401380">A Combinatorial proof of a partition identity of Andrews and Stanley</a>, International Journal of Mathematics and Mathematical Sciences, Volume 2004, Issue 47, Pages 2495-2501.
%H A097566 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A097566 R. P. Stanley, <a href="http://www.jstor.org/stable/3072410">Problem 10969</a>, Amer. Math. Monthly, 109 (2002), 760. as mentioned in link.
%F A097566 From _Michael Somos_, May 04 2011: (Start)
%F A097566 Expansion of q^(1/24) * eta(q^2)^2 * eta(q^16)^5 / (eta(q) * eta(q^4)^5 * eta(q^32)^2) in powers of q.
%F A097566 Expansion of phi(x^8) / (phi(x^2) * f(-x)) in powers of x where phi(), f() are Ramanujan theta functions.
%F A097566 Euler transform of period 32 sequence [ 1, -1, 1, 4, 1, -1, 1, 4, 1, -1, 1, 4, 1, -1, 1, -1, 1, -1, 1, 4, 1, -1, 1, 4, 1, -1, 1, 4, 1, -1, 1, 1, ...].
%F A097566 G.f.: theta_3(x^8) / (theta_3(x^2) * Product_{k>0} (1 - x^k)) = A000041(x) * A112128(x^2).
%F A097566 a(n) = (A000041(n) + A085261(n)) / 2.
%F A097566 (End)
%e A097566 G.f. = 1 + x + x^3 + 5*x^4 + 5*x^5 + x^6 + 5*x^7 + 20*x^8 + 20*x^9 + 6*x^10 + ...
%e A097566 G.f. = 1/q + q^23 + q^71 + 5*q^95 + 5*q^119 + q^143 + 5*q^167 + 20*q^191 + 20*q^215 + ...
%e A097566 a(5) = 5 because only the partitions {5}, {3,2}, {3,1,1}, {2,2,1}, {1,1,1,1,1} have conjugates resp. {1,1,1,1,1}, {2,2,1}, {3,1,1}, {3,2}, {5} with matching counts of odd elements (resp. (1,5), (1,1), (3,3), (1,1), (5,1) being congruent modulo 4 ).
%p A097566 with(combinat); t1:=mul( (1+q^(2*n-1))/((1-q^(4*n))*(1+q^(4*n-2))^2), n=1..100): t2:=series(t1,q,100): f:=n->coeff(t2,q,n); p:=numbpart; t:=n->(p(n)+f(n))/2; # _N. J. A. Sloane_, Jan 25 2009
%t A097566 fStanley[n_Integer]:=Product[(1+q^(2i-1))/(1-q^(4i))/(1+q^(4i-2))^2, {i, n}]; Table[PartitionsP[n]/2+1/2*Coefficient[Series[fStanley[n], {q, 0, n+1}], q^n], {n, 64}] or Table[Count[Partitions[n], q_/;Mod[Count[q, w_/;OddQ[w]]- Count[TransposePartition[q], w_/;OddQ[w]], 4]===0], {n, 24}]
%t A097566 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^8] / (EllipticTheta[ 3, 0, x^2] QPochhammer[ x]), {x, 0, n}]; (* _Michael Somos_, Jun 01 2014 *)
%o A097566 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^16 + A)^5 / (eta(x + A) * eta(x^4 + A)^5 * eta(x^32 + A)^2), n))}; /* _Michael Somos_, May 04 2011 */
%Y A097566 Cf. A000041, A085261, A097567, A112128, A190101.
%K A097566 easy,nonn
%O A097566 0,5
%A A097566 _Wouter Meeussen_, Aug 28 2004