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

1, 1, 0, 1, 5, 5, 1, 5, 20, 20, 6, 20, 65, 65, 25, 66, 185, 185, 85, 190, 481, 482, 250, 501, 1165, 1170, 666, 1230, 2666, 2685, 1646, 2850, 5827, 5887, 3830, 6303, 12251, 12415, 8487, 13395, 24912, 25323, 18052, 27507, 49215, 50176, 37072, 54832, 94781, 96905

Offset

0,5

Comments

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

Odd(p) is the number of odd parts of a partition p. a(n) is denoted t(n) in Problem 10969.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

George E. Andrews, On a Partition Function of Richard Stanley, The Electronic Journal of Combinatorics, Volume 11, Issue 2 (2004-6) (The Stanley Festschrift volume), Research Paper #R1.

M. Ishikawa and J. Zeng, The Andrews-Stanley partition function and Al-Salam-Chihara polynomials, 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.]

Andrew V. Sills, A Combinatorial proof of a partition identity of Andrews and Stanley, International Journal of Mathematics and Mathematical Sciences, Volume 2004, Issue 47, Pages 2495-2501.

Michael Somos, Introduction to Ramanujan theta functions

R. P. Stanley, Problem 10969, Amer. Math. Monthly, 109 (2002), 760. as mentioned in link.

Formula

From Michael Somos, May 04 2011: (Start)

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.

Expansion of phi(x^8) / (phi(x^2) * f(-x)) in powers of x where phi(), f() are Ramanujan theta functions.

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, ...].

G.f.: theta_3(x^8) / (theta_3(x^2) * Product_{k>0} (1 - x^k)) = A000041(x) * A112128(x^2).

a(n) = (A000041(n) + A085261(n)) / 2.

(End)

Example

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 + ...
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 + ...
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 ).

Maple

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

Mathematica

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}]
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^8] / (EllipticTheta[ 3, 0, x^2] QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, Jun 01 2014 *)

Prog

(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 */

Crossrefs

Cf. A000041, A085261, A097567, A112128, A190101.

Sequence in context: A198740 A082956 A190101 * A154945 A300710 A254347

Adjacent sequences: A097563 A097564 A097565 * A097567 A097568 A097569

Keyword

easy,nonn

Author

Wouter Meeussen, Aug 28 2004

Status

approved