A122190 -id:A122190 - cp's OEIS frontend

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

James Sellers, Feb 14 2000

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

			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 + ...

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

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.

Cf. A122190.

Cf. A000700, A000122, A010054, A121373.

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 *)

{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))}

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

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

{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))}

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

A094247 Expansion of (phi(-q^5)^2 - phi(-q)^2) / 4 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Apr 24 2004

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

			G.f. = q - q^2 - q^4 + q^5 - q^8 + q^9 - q^10 + 2*q^13 - q^16 + 2*q^17 - q^18 - q^20 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
S. Cooper, On Ramanujan's function k(q)=r(q)r^2(q^2), Ramanujan J., 20 (2009), 311-328; see p. 313 eq. (2.4).
L.-C. Shen, On the additive formulas of the theta functions and a collection of Lambert series pertaining to the modular equations of degree 5, Trans. Amer. Math. Soc. 345 (1994), no. 1, 323-345; see p. 336 eq. (3.13), p. 338 eq. (3.21).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A053694, A122190, A214316.

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q^5]^2 - EllipticTheta[ 4, 0, q]^2)/4, {q, 0, n}]; (* Michael Somos, Jul 12 2012 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ -q^5] QPochhammer[ q^20] QPochhammer[q, q^2], {q, 0, n}]; (* Michael Somos, Jul 12 2012 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^10 + A)^3 / (eta(x^2 + A) * eta(x^5 + A)),n))};

{a(n) = if( n<1, 0, -(-1)^n * sumdiv( n, d, kronecker( -100, d)))}; /* Michael Somos, Aug 24 2006 */

Expansion of q * f(q^5) * f(-q^20) * chi(-q) in powers of q where f() and chi() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^10)^3 / (eta(q^2) * eta(q^5)) in powers of q.
Euler transform of period 10 sequence [-1, 0, -1, 0, 0, 0, -1, 0, -1, -2, ...].
a(n) is multiplicative with a(2^e) = -1 if e > 0. a(5^e) = 1, a(p^e) = e+1 if p == 1, 5 (mod 8), a(p^e) = (1 + (-1)^e) / 2 if p == 3, 7 (mod 8).
G.f. is a period 1 Fourier series which satisfies f(-1 / (40 t)) = 4 (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A214316. - Michael Somos, Jul 12 2012
G.f.: x * Product_{k>0} (1 - x^k) * (1 - x^(10*k))^3 / ((1 - x^(2*k)) * (1 - x^(5*k))).
G.f.: Sum_{k>0} Kronecker( -100, k) * x^k / (1 + x^k) = Sum_{k>0} Kronecker( -25, k) * x^k * (1 - x^k)^2 / (1 - x^(4*k)). - Michael Somos, Jul 12 2012
a(n+1) = (-1)^n * A053694(n). a(4*n + 1) = A122190(n).
a(4*n + 3) = 0. a(2*n) = - A053694(n). - Michael Somos, Jul 12 2012

A133573 Expansion of ( 5 * phi(-q^5)^2 - phi(-q)^2 ) / 4 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, -1, 0, -1, -3, 0, 0, -1, 1, 3, 0, 0, 2, 0, 0, -1, 2, -1, 0, 3, 0, 0, 0, 0, -7, -2, 0, 0, 2, 0, 0, -1, 0, -2, 0, -1, 2, 0, 0, 3, 2, 0, 0, 0, -3, 0, 0, 0, 1, 7, 0, -2, 2, 0, 0, 0, 0, -2, 0, 0, 2, 0, 0, -1, -6, 0, 0, -2, 0, 0, 0, -1, 2, -2, 0, 0, 0, 0, 0, 3, 1, -2, 0, 0, -6, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 2, -1, 0, 7, 2, 0, 0, -2

Offset: 0

Michael Somos, Sep 17 2007

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

Multiplicative because this sequence is the inverse Moebius transform of a multiplicative sequence. - Andrew Howroyd, Aug 06 2018

			G.f. = 1 + q - q^2 - q^4 - 3*q^5 - q^8 + q^9 + 3*q^10 + 2*q^13 - q^16 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Shaun Cooper, On Ramanujan's function k(q)=r(q)r^2(q^2), Ramanujan J., 20 (2009), 311-328; see p. 313, eq. (2.5).
Li-Chien Shen, On the additive formulas of the theta functions and a collection of Lambert series pertaining to the modular equations of degree 5, Trans. Amer. Math. Soc. 345 (1994), no. 1, 323-345; see p. 335, eq. (3.5), p. 342, eq. (3.42).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A122190, A133574, A214316.

Cf. A000700, A000122, A010054, A121373.

a[ n_] := SeriesCoefficient[ (5 EllipticTheta[ 4, 0, q^5]^2 - EllipticTheta[ 4, 0, q]^2)/4, {q, 0, n}]; (* Michael Somos, Jul 12 2012 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^2 QPochhammer[ q^5, q^10] / QPochhammer[ q, q^2], {q, 0, n}]; (* Michael Somos, Jul 12 2012 *)

{a(n) = if( n<1, n==0, (-1)^n * sumdiv(n, d, if( d%5==0, kronecker(-4, d/5) * 5) - kronecker(-4, d)))};

{a(n) = local(A); if( n<0, 0, A = x*O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^5+A) / (eta(x + A) * eta(x^10 + A)), n))};

Expansion of eta(q^2)^3 * eta(q^5) / ( eta(q) * eta(q^10) ) in powers of q.
Euler transform of period 10 sequence [ 1, -2, 1, -2, 0, -2, 1, -2, 1, -2, ...].
Moebius transform is period 40 sequence [ 1, -2, -1, 0, -4, 2, -1, 0, 1, 8, -1, 0, 1, 2, 4, 0, 1, -2, -1, 0, 1, 2, -1, 0, -4, -2, -1, 0, 1, -8, -1, 0, 1, -2, 4, 0, 1, 2, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (40 t)) = 20 (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A122190.
a(n) = (-1)^n * A133574(n). a(2*n) = A133574(n). a(4*n + 1) = A214316(n). a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0. a(9*n) = a(n). - Michael Somos, Jul 12 2012
Sum_{k=1..n} abs(a(k)) ~ (8*Pi/25) * n. - Amiram Eldar, Jan 27 2024

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A053694 Number of self-conjugate 5-core partitions of n.

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A094247 Expansion of (phi(-q^5)^2 - phi(-q)^2) / 4 in powers of q where phi() is a Ramanujan theta function.

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A133573 Expansion of ( 5 * phi(-q^5)^2 - phi(-q)^2 ) / 4 in powers of q where phi() is a Ramanujan theta function.

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