A190101 Number of transpose partition pairs of order n whose number of odd parts differ by numbers of the form 4*k + 2.

0, 0, 1, 1, 0, 1, 5, 5, 1, 5, 18, 18, 6, 18, 55, 55, 23, 56, 150, 150, 73, 155, 376, 377, 205, 394, 885, 890, 526, 940, 1979, 1996, 1261, 2128, 4240, 4290, 2863, 4611, 8764, 8895, 6213, 9630, 17561, 17877, 12980, 19479, 34243, 34961, 26246, 38310, 65187

Offset: 0

Author

Michael Somos, May 04 2011

Keywords

• nonn

Comments

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

Denoted u(n)/2 by Lossers. t(n) is A097566(n). Stanley's f(n) is A085261(n). Partitions p(n) is A000041(n).

As noted in the solution the number of odd parts of a partition and its conjugate are of the same parity as n. Hence the difference in the number of odd parts must be even and if it is not divisible by 4 then it is of the form 4*k + 2 and the partition is not self conjugate.

Examples

			G.f. = x^2 + x^3 + x^5 + 5*x^6 + 5*x^7 + x^8 + 5*x^9 + 18*x^10 + 18*x^11 + ...
G.f. = q^47 + q^71 + q^119 + 5*q^143 + 5*q^167 + q^191 + 5*q^215 + ...
a(6) = 5 because ([6], [1,1,1,1,1,1]), ([5,1], [2,1,1,1,1]), ([4,2], [2,2,1,1]), ([4,1,1], [3,1,1,1]), ([3,3], [2,2,2]) are the 5 pairs of partitions of 6 where each partition and its transpose number of odd parts differ by 6, 2, 2, 2, 2 which are of the form 4*k + 2.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• O. P. Lossers, Comparing Odd Parts in Conjugate Partitions: Solution 10969, Amer. Math. Monthly, 111 (Jun-Jul 2004), pp. 536-539.
• R. P. Stanley, Problem 10969, Amer. Math. Monthly, 109 (2002), 760.

Crossrefs

Cf. A000041, A085261, A097566.

Programs

Mathematica

a[n_] := SeriesCoefficient[EllipticTheta[2, 0, q^8]/( 2*QPochhammer[q] * EllipticTheta[3, 0, q^2]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 18 2017 *)
nmax = 100; CoefficientList[Series[x^2 * Product[(1 + x^(4*k))^3 * (1 + x^(8*k)) * (1 + x^(16*k))^2 / ((1 + x^(2*k))^2 * (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 05 2025 *)

PARI

{a(n) = local(A); if( n<2, 0, n = n-2; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^8 + A) * eta(x^32 + A))^2 / (eta(x + A) * eta(x^4 + A)^5 * eta(x^16 + A)), n))};

Formula

Expansion of x^2 * psi(x^16) / (f(-x) * phi(x^2)) in powers of x where phi(), psi(), f() are Ramanujan theta functions.

Expansion of q^(1/24) * (eta(q^2) * eta(q^8) * eta(q^32))^2 / (eta(q) * eta(q^4)^5 * eta(q^16)) in powers of q.

Euler transform of period 32 sequence [ 1, -1, 1, 4, 1, -1, 1, 2, 1, -1, 1, 4, 1, -1, 1, 3, 1, -1, 1, 4, 1, -1, 1, 2, 1, -1, 1, 4, 1, -1, 1, 1, ...].

p(n) = t(n) + u(n). f(n) = t(n) - u(n). u(n) = 2*a(n).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (16*n*sqrt(3)). - Vaclav Kotesovec, Jul 05 2025