cp's OEIS Frontend

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

%I A190101 #18 Jul 05 2025 06:50:02
%S A190101 0,0,1,1,0,1,5,5,1,5,18,18,6,18,55,55,23,56,150,150,73,155,376,377,
%T A190101 205,394,885,890,526,940,1979,1996,1261,2128,4240,4290,2863,4611,8764,
%U A190101 8895,6213,9630,17561,17877,12980,19479,34243,34961,26246,38310,65187
%N A190101 Number of transpose partition pairs of order n whose number of odd parts differ by numbers of the form 4*k + 2.
%C A190101 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A190101 Denoted u(n)/2 by Lossers. t(n) is A097566(n). Stanley's f(n) is A085261(n). Partitions p(n) is A000041(n).
%C A190101 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.
%H A190101 G. C. Greubel, <a href="/A190101/b190101.txt">Table of n, a(n) for n = 0..1000</a>
%H A190101 O. P. Lossers, <a href="http://www.jstor.org/stable/4145085">Comparing Odd Parts in Conjugate Partitions: Solution 10969</a>, Amer. Math. Monthly, 111 (Jun-Jul 2004), pp. 536-539.
%H A190101 R. P. Stanley, <a href="http://www.jstor.org/stable/3072410">Problem 10969</a>, Amer. Math. Monthly, 109 (2002), 760.
%F A190101 Expansion of x^2 * psi(x^16) / (f(-x) * phi(x^2)) in powers of x where phi(), psi(), f() are Ramanujan theta functions.
%F A190101 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.
%F A190101 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, ...].
%F A190101 p(n) = t(n) + u(n). f(n) = t(n) - u(n). u(n) = 2*a(n).
%F A190101 a(n) ~ exp(Pi*sqrt(2*n/3)) / (16*n*sqrt(3)). - _Vaclav Kotesovec_, Jul 05 2025
%e A190101 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 + ...
%e A190101 G.f. = q^47 + q^71 + q^119 + 5*q^143 + 5*q^167 + q^191 + 5*q^215 + ...
%e A190101 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.
%t A190101 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 *)
%t A190101 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 *)
%o A190101 (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))};
%Y A190101 Cf. A000041, A085261, A097566.
%K A190101 nonn
%O A190101 0,7
%A A190101 _Michael Somos_, May 04 2011