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

0,13

Comments

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

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.

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

Example

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 + ...

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 *)

Prog

(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))}

Crossrefs

Cf. A122190.

Cf. A000700, A000122, A010054, A121373.

Sequence in context: A059882 A303206 A094247 * A085862 A257392 A237123

Adjacent sequences: A053691 A053692 A053693 * A053695 A053696 A053697

Keyword

easy,nice,nonn

Author

James A. Sellers, Feb 14 2000

Status

approved