A053694 Number of self-conjugate 5-core partitions of n.
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, 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, 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
Offset: 0
Examples
1 + x + x^3 + x^4 + x^7 + x^8 + x^9 + 2*x^12 + x^15 + 2*x^16 + x^17 + ... q + q^2 + q^4 + q^5 + q^8 + q^9 + q^10 + 2*q^13 + q^16 + 2*q^17 + q^18 + ...
References
- Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 258, Entry 9(iii).
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Frank Garvan, Dongsu Kim, and Dennis Stanton, Cranks and t-cores, Invent. Math. 101 (1990), no. 1, 1-17.
- Christopher R. H. Hanusa and Rishi Nath, The number of self-conjugate core partitions, arxiv:1201.6629 [math.NT], 2012. See Table 1, p. 15.
- Michael Somos, Introduction to Ramanujan theta functions, 2019.
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ (EllipticTheta[3, 0, q]^2 - EllipticTheta[3, 0, q^5]^2) / (4 q), {q, 0, n}] (* Michael Somos, Jul 11 2011 *) 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 *) -
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))} -
PARI
{a(n) = if( n<0, 0, n++; sumdiv( n, d, kronecker( -100, d)))} -
PARI
{a(n) = if( n<0, 0, n++; direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( -100, p) * X))[n])} -
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))}
Formula
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)
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).
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.
From Michael Somos, Apr 25 2003: (Start)
Expansion of q^(-1) * eta(q^2)^2 * eta(q^5) * eta(q^20) / (eta(q) * eta(q^4)) in powers of q.
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, ...].
G.f.: Product_{k>0} (1 - x^(10*k))^2 * (1 + x^(2*k - 1)) / (1 + x^(10*k - 5)). (End)
a(4*n) = A122190(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/5. - Amiram Eldar, Jan 27 2024
Comments