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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 2 07:33 EST 2021. Contains 349437 sequences. (Running on oeis4.)