%I #36 Jan 27 2024 07:08:59

%S 1,1,0,1,1,0,0,1,1,1,0,0,2,0,0,1,2,1,0,1,0,0,0,0,1,2,0,0,2,0,0,1,0,2,

%T 0,1,2,0,0,1,2,0,0,0,1,0,0,0,1,1,0,2,2,0,0,0,0,2,0,0,2,0,0,1,2,0,0,2,

%U 0,0,0,1,2,2,0,0,0,0,0,1,1,2,0,0,2,0,0,0,2,1,0,0,0,0,0,0,2,1,0,1,2,0,0,2,0

%N Number of self-conjugate 5-core partitions of n.

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

%D Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 258, Entry 9(iii).

%H G. C. Greubel, <a href="/A053694/b053694.txt">Table of n, a(n) for n = 0..1000</a>

%H Frank Garvan, Dongsu Kim, and Dennis Stanton, <a href="http://dx.doi.org/10.1007/BF01231493">Cranks and t-cores</a>, Invent. Math. 101 (1990), no. 1, 1-17.

%H Christopher R. H. Hanusa and Rishi Nath, <a href="https://arxiv.org/abs/1201.6629">The number of self-conjugate core partitions</a>, arxiv:1201.6629 [math.NT], 2012. See Table 1, p. 15.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.

%F G.f.: product((1-q^(10*i))^2*(1-q^(10*i-5))*(1-q^(4*i-2))/((1-q^(2*i-1))*(1-q^(20*i-10))), i=1..200)

%F a(n) = b(n + 1) where b(n) is multiplicative and b(2^e) = b(5^e) = 1, b(p^e) = e+1 if p == 1, 5 (mod 8), b(p^e) = (1+(-1)^e)/2 if p == 3, 7 (mod 8).

%F Expansion of (phi(x)^2 - phi(x^5)^2) / (4*x) = chi(x) * f(-x^5) * f(-x^20) in powers of x where phi(), chi(), f() are Ramanujan theta functions.

%F From _Michael Somos_, Apr 25 2003: (Start)

%F Expansion of q^(-1) * eta(q^2)^2 * eta(q^5) * eta(q^20) / (eta(q) * eta(q^4)) in powers of q.

%F Euler transform of period 20 sequence [1, -1, 1, 0, 0, -1, 1, 0, 1, -2, 1, 0, 1, -1, 0, 0, 1, -1, 1, -2, ...].

%F G.f.: Product_{k>0} (1 - x^(10*k))^2 * (1 + x^(2*k - 1)) / (1 + x^(10*k - 5)). (End)

%F a(4*n) = A122190(n).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/5. - _Amiram Eldar_, Jan 27 2024

%e 1 + x + x^3 + x^4 + x^7 + x^8 + x^9 + 2*x^12 + x^15 + 2*x^16 + x^17 + ...

%e q + q^2 + q^4 + q^5 + q^8 + q^9 + q^10 + 2*q^13 + q^16 + 2*q^17 + q^18 + ...

%t a[ n_] := SeriesCoefficient[ (EllipticTheta[3, 0, q]^2 - EllipticTheta[3, 0, q^5]^2) / (4 q), {q, 0, n}] (* _Michael Somos_, Jul 11 2011 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[-q, q^2] QPochhammer[q^5, q^5] QPochhammer[q^20, q^20], {q, 0, n}] (* _Michael Somos_, Jul 11 2011 *)

%o (PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=0, n\2, 1 + x^(2*k + 1), 1 + x * O(x^n)) * prod( k=0, n\10, (1 - x^(10*k + 10))^2 / (1 + x^(10*k + 5)), 1 + x*O(x^n)), n))}

%o (PARI) {a(n) = if( n<0, 0, n++; sumdiv( n, d, kronecker( -100, d)))}

%o (PARI) {a(n) = if( n<0, 0, n++; direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( -100, p) * X))[n])}

%o (PARI) {a(n) = local(A); if(n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^5 + A) * eta(x^20 + A) / eta(x + A) / eta(x^4 + A), n))}

%Y Cf. A122190.

%Y Cf. A000700, A000122, A010054, A121373.

%K easy,nice,nonn

%O 0,13

%A _James A. Sellers_, Feb 14 2000