login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A053692 Number of self-conjugate 4-core partitions of n. 21
1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 2, 0, 1, 1, 1, 2, 0, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 2, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 3, 1, 0, 1, 0, 2, 1, 0, 1, 1, 1, 0, 1, 0, 0, 2, 0, 1, 0, 1, 2, 2, 0, 1, 0, 0, 2, 1, 1, 1, 2, 0, 0, 0, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 1, 2, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,11

COMMENTS

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

Also the number of positive odd solutions to equation x^2 + 4*y^2 = 8*n + 5. - Seiichi Manyama, May 28 2017

REFERENCES

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 153 Entry 22.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000

F. Garvan, D. Kim and D. Stanton, Cranks and t-cores, Inventiones Math. 101 (1990) 1-17.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of psi(x) * psi(x^4) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Nov 03 2005

Expansion of chi(x) * f(-x^8)^2 in powers of x where chi(), f() are Ramanujan theta functions. - Michael Somos, Jul 24 2012

Expansion of f(x, x^7) * f(x^3, x^5) = f(x, x^3) * f(x^4, x^12) in powers of x where f(,) is the Ramanujan general theta function. - Michael Somos, Jun 21 2015

Expansion of (psi(x)^2 - psi(-x)^2) / (4*x) in powers of x^2 where psi() is a Ramanujan theta function. - Michael Somos, Jun 21 2015

Expansion of q^(-5/8) * eta(q^2)^2 * eta(q^8)^2 / (eta(q) * eta(q^4)) in powers of q. - Michael Somos, Apr 28 2003

Euler transform of period 8 sequence [ 1, -1, 1, 0, 1, -1, 1, -2, ...]. - Michael Somos, Apr 28 2003

a(n) = 1/2 * b(8*n + 5), where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 4). - Michael Somos, Jul 24 2012

G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246950.

G.f.: Sum_{k in Z} x^k / (1 - x^(8*k + 5)). - Michael Somos, Nov 03 2005

G.f.: Sum_{k>0} -(-1)^k * x^((k^2 + k)/2) / (1 - x^(2*k - 1)). - Michael Somos, Jun 21 2015

G.f.: product((1-x^(8*i))^2*(1-x^(4*i-2))/(1-x^(2*i-1)), i=1..infinity)

a(9*n + 2) = a(9*n + 8) = 0. a(9*n + 5) = a(n). 2 * a(n) = A008441(2*n + 1).

EXAMPLE

G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + x^7 + 2*x^10 + x^12 + x^13 + x^14 + 2*x^15 + ...

G.f. = q^5 + q^13 + q^29 + q^37 + q^45 + q^53 + q^61 + 2*q^85 + q^101 + q^109 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x (1/2)] EllipticTheta[ 2, 0, x^2] / (4 x^(5/8)), {x, 0, n}]; (* Michael Somos, Jun 21 2015 *)

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ x^8]^2, {x, 0, n}]; (* Michael Somos, Jun 21 2015 *)

a[ n_] := SeriesCoefficient[ QPochhammer[ x^8]^2 QPochhammer[ x^2, x^4] / QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Jun 21 2015 *)

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^2]^2 - 2 EllipticTheta[ 2, Pi/4, q^2]^2) / 16, {q, 0, 8 n + 5}]; (* Michael Somos, Jun 21 2015 *)

a[ n_] := If[ n < 0, 0, Sum[ (-1)^Quotient[d, 2], {d, Divisors[ 8 n + 5]}] / 2]; (* Michael Somos, Jun 21 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = sum( k=0, ceil( sqrtint(8*n + 1)/2), x^((k^2 + k)/2), x * O(x^n)); polcoeff( A * subst(A + x * O(x^(n\4)), x, x^4), n))}; /* Michael Somos, Nov 03 2005 */

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^8 + A)^2 / (eta(x + A) * eta(x^4 + A)), n))}; /* Michael Somos, Apr 28 2003 */

(PARI) {a(n) = if( n<0, 0, sumdiv( 8*n + 5, d, (-1)^(d\2)) / 2)}; /* Michael Somos, Jun 21 2015*/

(MAGMA) A := Basis( ModularForms( Gamma1(64), 1), 701); A[6] + A[14] + A[30] - A[35] + A[36]; /* Michael Somos, Jun 21 2015 */;

CROSSREFS

Cf. A008441, A053693.

Sequence in context: A272903 A321458 A226194 * A341024 A286934 A282714

Adjacent sequences:  A053689 A053690 A053691 * A053693 A053694 A053695

KEYWORD

easy,nonn

AUTHOR

James A. Sellers, Feb 14 2000

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 17 22:38 EDT 2021. Contains 347489 sequences. (Running on oeis4.)